diff --git a/stable b/stable index f4caccc..d34dcac 120000 --- a/stable +++ b/stable @@ -1 +1 @@ -v0.1.2 \ No newline at end of file +v0.1.3 \ No newline at end of file diff --git a/v0.1 b/v0.1 index f4caccc..d34dcac 120000 --- a/v0.1 +++ b/v0.1 @@ -1 +1 @@ -v0.1.2 \ No newline at end of file +v0.1.3 \ No newline at end of file diff --git a/v0.1.3/.documenter-siteinfo.json b/v0.1.3/.documenter-siteinfo.json new file mode 100644 index 0000000..9d2c83b --- /dev/null +++ b/v0.1.3/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.8.5","generation_timestamp":"2023-12-06T15:48:39","documenter_version":"1.2.1"}} \ No newline at end of file diff --git a/v0.1.3/BellScenario/combinatorics/index.html b/v0.1.3/BellScenario/combinatorics/index.html new file mode 100644 index 0000000..8a5bd23 --- /dev/null +++ b/v0.1.3/BellScenario/combinatorics/index.html @@ -0,0 +1,36 @@ + +Combinatorics · BellScenario.jl

Combinatorics Utilities

Set Partitions

BellScenario.stirling2 — Function
stirling2( n :: Int64, k :: Int64  ) :: Int64

Counts the number of ways to partition n items into k unlabelled groups. This quantity is known as Stirling's number of the 2nd kind:

\[\left\{n \atop k \right\} = \frac{1}{k!}\sum_{i=0}^k (-1)^i\binom{k}{i}(k-i)^n\]

Throws a DomainError if inputs do not satisfy n ≥ k ≥ 1.

source
BellScenario.stirling2_partitions — Function
stirling2_partitions( n :: Int64, k :: Int64 ) :: Vector{Vector{Vector{Int64}}}

Enumerates the unique partitions of n items into k unlabelled sets. Each partition is a vector containing a set of k vectors designating each group.

E.g.

julia> stirling2_partitions(4, 2)
+7-element Vector{Vector{Vector{Int64}}}:
+ [[1, 2, 3], [4]]
+ [[3], [1, 2, 4]]
+ [[1, 2], [3, 4]]
+ [[1, 3], [2, 4]]
+ [[2], [1, 3, 4]]
+ [[2, 3], [1, 4]]
+ [[1], [2, 3, 4]]

This recursive algorithm was inspired by this blog.

source
BellScenario.stirling2_matrices — Function
stirling2_matrices( n :: Int64, k :: Int64 ) :: Vector{Matrix{Bool}}

Generates the set of matrices with k rows and n columns where rows correspond to the groups and columns are the grouped elements. A non-zero element designates that the column id is grouped into the corresponding row.

E.g.

julia> stirling2_matrices(4, 2)
+7-element Vector{Matrix{Bool}}:
+ [1 1 1 0; 0 0 0 1]
+ [0 0 1 0; 1 1 0 1]
+ [1 1 0 0; 0 0 1 1]
+ [1 0 1 0; 0 1 0 1]
+ [0 1 0 0; 1 0 1 1]
+ [0 1 1 0; 1 0 0 1]
+ [1 0 0 0; 0 1 1 1]

A DomainError is thrown if n ≥ k ≥ 1 is not satisfied.

source

Matrix Constructions

BellScenario.permutation_matrices — Function
permutation_matrices( dim :: Int64 ) :: Vector{Matrix{Bool}}

Generates the set of square permutation matrices of dimension dim.

E.g.

julia> permutation_matrices(3)
+6-element Vector{Matrix{Bool}}:
+ [1 0 0; 0 1 0; 0 0 1]
+ [1 0 0; 0 0 1; 0 1 0]
+ [0 1 0; 1 0 0; 0 0 1]
+ [0 0 1; 1 0 0; 0 1 0]
+ [0 1 0; 0 0 1; 1 0 0]
+ [0 0 1; 0 1 0; 1 0 0]
source
BellScenario.n_choose_k_matrices — Function
n_choose_k_matrices( n :: Int64, k :: Int64 ) :: Vector{Matrix{Bool}}

Generates a set of n by k matrices which represent all combinations of selecting k columns from n rows. Each column, contains a single non-zero element and k rows contain a non-zero element.

E.g.

julia> n_choose_k_matrices( 4, 2 )
+6-element Vector{Matrix{Bool}}:
+ [1 0; 0 1; 0 0; 0 0]
+ [1 0; 0 0; 0 1; 0 0]
+ [1 0; 0 0; 0 0; 0 1]
+ [0 0; 1 0; 0 1; 0 0]
+ [0 0; 1 0; 0 0; 0 1]
+ [0 0; 0 0; 1 0; 0 1]

A DomainError is thrown if n ≥ k ≥ 1 is not satisfied.

source

Alternative Number Bases

BellScenario.base_n_val — Function
base_n_val(
+    num_array :: Vector{Int64},
+    base :: Int64;
+    big_endian=true :: Bool
+) :: Int64

Given an array representing a number in base-n returns the value of that number in base-10.

Inputs:

  • num_array - Vector containing semi-positive integers less than base.
  • base - The base-n number represented by num_array.
  • big_endian - true if most significant place is at index 1, else false.
source
diff --git a/v0.1.3/BellScenario/games/index.html b/v0.1.3/BellScenario/games/index.html new file mode 100644 index 0000000..d27ba59 --- /dev/null +++ b/v0.1.3/BellScenario/games/index.html @@ -0,0 +1,21 @@ + +Games · BellScenario.jl

BellScenario.jl - Games

BellScenario.AbstractGame — Type

An AbstractGame is the abstract type that is parent to type representing a cost function for strategies. A Game is a linear inequality dual to Strategy matrices, it is represented by a matrix $G$ containing scalar coefficients and a bound $\beta$. The bound of the linear inequality is typically a maximum score attainable by Bell scenario using a certain set of resources. A Game $G$ is played by a Strategy $S$ to achieve a score computed as,

\[\beta \geq \langle G, S\rangle = \sum_{x,y} G_{y,x}S(y|x).\]

The game is "won" if the strategy scores greater than $\beta$, that is, the above inequality is violated. Since $\beta$ is the maximum score for a set of resources, the game is won only if the tested strategy used a set of resources of greater operational value than than considered when computing the bound $\beta$.

Conveniently, Strategy matrices have normalized columns and non-negative elements. This means that any game inequality can be converted into a form where game matrix $G$ has positive elements and $\beta$ designates a positive upper bound.

source
BellScenario.Game — Type
Game(game::Matrix{T}, β::Real) <: AbstractGame{T}

A Game is represented by a Matrix of coefficients and a scalar bound β.

Type parameter T can be either Int64 or Float64.

source
BellScenario.BellGame — Type
BellGame(game::Matrix{Int64}, β::Int64, scenario::Scenario)

A BellGame represents a Bell inequality or tight facet of the local polytope. Since the vertices of the local polytope are deterministic strategies with 0,1 elements, the linear inequalities describing facets of the local polytope have rational coefficients. Therefore, if a inequality tightly bounds the local polytope, it can be represented by a game with integer coefficients.

source

Conversion Methods

A BellGame is a matrix representation of a linear inequality that bounds a LocalPolytope for some Scenario. For convenience, conversion methods are provided to transform BellGames to alternative representations of polytope facets. There are two supported types which can be converted to/from BellGames:

  • Facet - Vector{Int64}, a column-major vectorization of the game matrix with the bound placed in the last element of the vector.
  • IEQ - A facet data structure defined in XPORTA.jl.
Base.convert — Method

Facet (Vector{Int64}) -> BellGame

convert(
+    ::Type{BellGame},
+    facet::Vector{Int64},
+    scenario::Union{BlackBox, LocalSignaling};
+    rep = "normalized"::String
+)
source
Base.convert — Method

Facet (Vector{Int64}) -> BellGame

convert(
+    ::Type{BellGame},
+    facet::Vector{Int64},
+    scenario::BipartiteNonSignaling;
+    rep = "non-signaling"::String
+)

Transforms LocalPolytope facets into BellGame types.

source
Base.convert — Method

BellGame -> Facet (Vector{Int64})

convert(::Type{Vector{Int64}}, BG::BellGame; rep = "normalized" :: String)
source
Base.convert — Method

BellGame -> Vector{Int64}

convert(::Type{Vector{Int64}},
+    BG::BellGame,
+    scenario::BipartiteNonSignaling;
+    rep = "non-signaling" :: String
+)

Transforms a BellGame for a BipartiteNonSignaling scenario into a facet vector.

source
Base.convert — Method

XPORTA.IEQ to BellGame's

convert(
+    ::Type{Vector{BellGame}},
+    ieq::IEQ,
+    scenario::Union{BlackBox, LocalSignaling};
+    rep = "normalized" :: String
+)
source
Base.convert — Method

BellGame's to XPORTA.IEQ

convert(::Type{IEQ}, bell_games::Vector{BellGame}; rep = "normalized" :: String)
source

File I/O

In practice, one may need to view are large set of BellGames in a human-readable form.

BellScenario.pretty_print_txt — Function
pretty_print_txt( bell_games :: Vector{BellGame}, filename :: String )

Prints a set of BellGame's to filename.txt in a human-readable form.

source
diff --git a/v0.1.3/BellScenario/overview/index.html b/v0.1.3/BellScenario/overview/index.html new file mode 100644 index 0000000..aa4bc3a --- /dev/null +++ b/v0.1.3/BellScenario/overview/index.html @@ -0,0 +1,2 @@ + +Overview · BellScenario.jl

BellScenario.jl - Overview

BellScenario — Module

Types and constructors that represent Bell scenarios, their statistics, and their bounds.

The BellScenario.jl module is the base library for Bell scenario analysis.

Features:

A game theoretic framework is used to evaluate the performance of black-box systems. A cost function testing the black-box system is regarded as a game while the statistics generated by the black-box system are regarded as a strategy. A strategy is played against a game to achieve a score, hence, this framework provides a quantitative metric of the performance of a black-box system in regards to a particular task.

source

Black-Box Devices

A black-box device describes a physical system with inputs and outputs that an observer can test, however, the observer has no knowledge of the physical system contained within the device.

Black-Box Device

A black-box device accepts and input $x$ and produces an output $y$. No assumptions are made about how $y$ is computed from $x$. A black-box is characterized by its conditional probability distribution $S(y|x)$ which can be derived by collecting the input-output statistics of the black-box. The distribution $S(y|x)$ is a stochastic map taking input $x$ to output $y$. In BellScenario.jl, stochastic maps are referred to as strategies and outlined in greater detail in the BellScenario.jl - Strategies section.

Resources

The statistics achieved by Bell scenario depend on the resources available in the black-box system. In the context of Bell scenarios, resources describe the connection between black-boxes. Resources are typically limited and place restrictions on the black-box system.

Communication Resources:

Two (or more) black-boxes may coordinate using a limited amount of communication.

  • Classical Communication - A classical state is sent from one black-box to another.
  • Quantum Communication - A quantum state is sent from one black-box to another.

Communication resources are used during a Bell test. Classical communication resources are measured in dits which describes the number of distinct messages that can be sent. Quantum communication resources are measured in qudits which describes the Hilbert space dimension of the quantum state used to communicate. Quantum communication can simulate classical communication by encoding the classical messages as an orthogonal basis on the qudit Hilbert space.

Correlation Resources:

Two (or more) black-boxes may coordinate using a limited amount of correlation.

  • Shared Randomness - Two or more black-boxes share a random variable $\lambda$ drawn from a sample space $\Lambda$ with probability $q(\lambda)$.
  • Quantum Entanglement - Two or more black-boxes share a non-separable quantum state.

Correlation resources may be distributed before any Bell tests are run.

Diagrams

A Bell scenario is depicted by a Directed Acyclic Graph (DAG) which shows the causal flow of information through the black-box system. For example, the following figure shows a bipartite signaling scenario.

Bipartite Signaling Scenario Diagram

In the diagram, the information flows from inputs $x$ and $y$ to outputs $a$ and $b$. In general, a free origin of an arrow corresponds to an input of the Bell scenario while a free point of an arrow corresponds to an output of the Bell scenario. The blue rectangular nodes in the graph represent black-box devices, e.g. $A_\lambda(a|x)$ and $B_\lambda(b|m,y)$. When communication is present between two black-boxes, it is depicted by a solid arrow originating at one node and terminating at another. The red circular nodes in the graph, e.g. $\Lambda$, represent correlation resources. When a correlation resource is shared between two black-boxes, it is depicted by a red dotted arrow flowing from the correlation resource node to the each of the black-box nodes.

diff --git a/v0.1.3/BellScenario/scenarios/index.html b/v0.1.3/BellScenario/scenarios/index.html new file mode 100644 index 0000000..4b3c50d --- /dev/null +++ b/v0.1.3/BellScenario/scenarios/index.html @@ -0,0 +1,16 @@ + +Scenarios · BellScenario.jl

BellScenario.jl - Scenarios

BellScenario.Scenario — Type

A Scenario is an abstract type parent to all Bell scenarios. Each child of this abstract type describes a distinct black-box system configuration.

source
BellScenario.BlackBox — Type
BlackBox(num_out :: Int64, num_in :: Int64) <: Scenario

A Bell scenario consisting of a single black-box device with num_in inputs and num_out outputs.

Black-Box Device

The black-box device computes the output $y$ from the input $x$ by performing a stochastic map $S(y|x)$.

Errors

A DomainError is thrown if parameters num_out or num_in is less than 1.

source
BellScenario.LocalSignaling — Type
LocalSignaling(
+    X :: Int64,
+    Y :: Int64,
+    d :: Int64,
+) <: Scenario

A bipartite signaling scenario where information is passed from a transmitter black-box to a receiver black-box using no more than d dits of communication.

Local Signaling Scenario

The transmitter device has X inputs and the receiver device has Y outputs and shared randomness is held between the two devices. When quantum communication is used instead of classical communication no Bell violations occur.

Errors

A DomainError is thrown if X, Y, or d is less than 1.

source
BellScenario.BipartiteNonSignaling — Type
BipartiteNonSignaling(
+    A :: Int64,
+    B :: Int64,
+    X :: Int64,
+    Y :: Int64
+) <: Scenario

A bipartite non-signaling scenario where each device receives an input and produces an output. Let Alice be the device with A outputs and X inputs while Bob is the device with B outputs and Y inputs.

Bipartite Non-Signaling Scenario

Shared randomness is held between Alice and Bob. When Alice and Bob share quantum entanglement, Bell violations are known to occur.

Errors

A DomainError is thrown if A, B, X, or Y is less than 1.

source
BellScenario.BipartiteSignaling — Type
BipartiteSignaling(
+    A :: Tuple{Int64, Int64},
+    B :: Tuple{Int64, Int64};
+    dits :: Int64 = 1,
+    bidirectional :: Bool = false
+) <: Scenario

A bipartite signaling scenario where each device can send a message to the other.

Bipartite Signaling Scenario

source
diff --git a/v0.1.3/BellScenario/strategies/index.html b/v0.1.3/BellScenario/strategies/index.html new file mode 100644 index 0000000..42da70f --- /dev/null +++ b/v0.1.3/BellScenario/strategies/index.html @@ -0,0 +1,40 @@ + +Strategies · BellScenario.jl

BellScenario.jl - Strategies

BellScenario.AbstractStrategy — Type

An AbstractStrategy is an abstract type parent to all strategy matrices. A black-box device is characterized by its conditional probabilities which can be organized into a strategy matrix or column-stochastic map $S : \mathcal{X} \to \mathcal{Y}$. The strategy matrix $S$ is constructed as follows,

\[S = \sum_{x,y} P(y|x) |y\rangle\langle x|\]

where the elements of a strategy matrix must be non-negative and normalized:

  • $P(y|x) \geq 0$
  • $\sum_y P(y|x) = 1$

Since strategies are just stochastic matrices, the product of two strategies yields a new strategy, e.g. $S_A*S_B = S_C$.

*(S1::AbstractStrategy, S2::AbstractStrategy) :: Strategy

When multiplied together, two strategies may represent a new Scenario, hence, a the multiplication operator can also be passed a Scenario.

*(S1::AbstractStrategy, S2::AbstractStrategy, scenario::Scenario) :: Strategy
source
BellScenario.Strategy — Type
Strategy(conditionals :: Matrix{<:Real}; atol::Float64 = 1e-7) <: AbstractMatrix{Float64}
+
+Strategy(conditionals :: Conditionals; atol::Float64 = 1e-7) <: AbstractMatrix{Float64}

The conditionals parameter is a column stochastic matrix. The conditionals can either be accepted in a raw Matrix{<:Real} format or as a Conditionals type from QBase.jl. The atol parameter expresses the absolute tolerance for numerical error in the constraints of conditional probablities. By default, the constructor creates a strategy for a 'BlackBox' scenario. However, a Scenario can be passed to the Strategy constructor.

Strategy(conditionals :: Matrix{<:Real}, scenario :: Scenario, atol::Float64=1e-7)

Errors:

A DomainError is thrown if:

  • The provided Scenario does not match the expected dimension of the conditionals matrix.
  • The conditionals matrix is not a valid stochastic matrix, e.g. non-negative and normalized.
source
BellScenario.strategy_dims — Function
strategy_dims( scenario :: Scenario ) :: Tuple{Int64, Int64}

Returns the dimensions of the Conditionals matrix describing a Strategy for the Scenario at hand. Each Scenario, can place unique constraints on the matrix dimensions, therfore, separate methods are called for each concrete Scenario.

source
BellScenario.random_strategy — Function
random_strategy(
+    num_inputs :: Int64,
+    num_outputs :: Int64;
+) :: Strategy

Constructs a randomized Strategy matrix. Zeros are inserted into the strategy to ensure that it does not closely resemble a uniform distribution.

source

Deterministic Strategies

BellScenario.DeterministicStrategy — Type
DeterministicStrategy(conditionals :: Matrix{Int64}) <: AbstractStrategy{Int64}

A strategy matrix describing the behavior of a deterministic black-box. A strategy deterministic if its elements satisfy $P(y|x)\in\{0,1\}$ in addition to the non-negativity and normalization constraints. By default, the constructor creates a strategy for a 'BlackBox' scenario, however, a Scenario can be passed to the DeterministicStrategy constructor.

DeterministicStrategy(conditionals :: Matrix{Int}, scenario :: Scenario)

The product of two deterministic strategies is a DeterministicStrategy.

*(S1::DeterministicStrategy, S2::DeterministicStrategy) :: DeterministicStrategy

When multiplied a new Scenario can be specified.

*(
+    S1::DeterministicStrategy,
+    S2::DeterministicStrategy,
+    scenario::Scenario
+) :: Deterministic Strategy

Errors:

A DomainError is thrown if:

  • The provided Scenario does not match the dimension of the conditionals matrix.
  • The elements of conditionals are not 0 or 1.
  • The strategy elements are not non-negative and normalized.
source
BellScenario.is_deterministic — Function
is_deterministic( strategy :: AbstractMatrix  ) :: Bool

Returns true if all elements of strategy are either 0 or 1 and the matrix is a valid conditional probability distribution.

source
BellScenario.deterministic_strategies — Function
deterministic_strategies(scenario :: BlackBox) :: Vector{Matrix{Int64}}
+
+deterministic_strategies(num_out :: Int64, num_in :: Int64) :: Vector{Matrix{Int64}}

Enumerates the set of deterministic strategies for the specified BlackBox. For performance, enumerated deterministic strategies are left as matrices and not constructed into DeterministicStrategy types.

source

Quantum Strategies

BellScenario.quantum_strategy — Function

Constructs a strategy matrix given quantum states and measurements. The supported scenarios include:

BlackBox scenarios:

quantum_strategy(
+    Π :: AbstractVector{<:AbstractMatrix},
+    ρ_states :: Vector{<:State};
+    atol=1e-7::Float64
+) :: Strategy

For a quantum system the conditional proabilities are constructed as

\[ P(y|x) = \text{Tr}[\Pi_y\rho_x].\]

source

LocalSignaling scenarios:

quantum_strategy(
+    Π :: AbstractVector{<:AbstractMatrix},
+    ρ_states :: Vector{<:AbstractMatrix},
+    scenario :: LocalSignaling;
+    atol=1e-7::Float64
+) :: Strategy

For quantum systems the conditional probabilities are construct as

\[ P(y|x) = \text{Tr}[\Pi_y\rho_x].\]

A DomainError is thrown if the provided states and measurements are not compatible with the specified scenario.

source

BipartiteNonSignaling scenarios:

quantum_strategy(
+        ρ_AB :: AbstractMatrix,
+        Π_Ax :: Vector{<:AbstractVector{<:AbstractMatrix}},
+        Π_By :: Vector{<:AbstractVector{<:AbstractMatrix}},
+        scenario :: BipartiteNonSignaling;
+        atol::Float64 = 1e-7
+)

Constructs a strategy matrix in the generalized representation for the quantum system with conditional probabilities.

\[ P(ab|xy) = \text{Tr}[(\Pi_a^x\otimes\Pi_b^y)\rho_{AB}]\]

A DomainError is thrown if

  • The length of each POVM does not match the scenarios number of outputs
  • The number of each party's POVMS doesn't match the the number of inputs.
source

Behaviors

The conditional probabilities of a Bell scenario can be represented by data structures other than a Strategy matrix. Most notably, the conditional probabilities can be organized into a vector referred to as a behavior in the literature. A behavior is a column-major vectorization of a strategy matrix e.g. S[:]. The dimension of the behavior vector (or strategy matrix) can be reduced to an equivalent subspace by using the normalization and non-signaling constraints to removed redundant parameters of the conditional probability distribution.

Subspaces used within BellScenario.jl include:

  • "generalized" - All conditional probabilities $P(y|x)$ for the Bell scenario are present.
  • "normalized" - The normalization constraint on each column is used to remove the last row of the strategy matrix.
  • "non-signaling" - The non-signaling constraints are applied to reduce the "normalized" subspace further.

Conversion Methods

Strategy matrices and behavior vectors are isomorphic representations. Therefore, a behavior can be converted into a strategy and vice versa.

Currently, behaviors are loosely represented by Vector{Int64} and Vector{Float64} types and a rep argument is used to specify the subspace. This may change in future updates of BellScenario.jl.

Base.convert — Method

Vertex (Vector{Int64}) -> DeterministicStrategy

convert(
+    ::Type{DeterministicStrategy},
+    vertex  :: Vector{Int64},
+    scenario :: Scenario;
+    rep = "normalized" :: String
+)
source
Base.convert — Method
convert(
+    S :: Type{<:AbstractStrategy}, vertex::Vector{<:Real}, scenario::BipartiteNonSignaling;
+    rep="non-signaling" :: String
+)

Transforms a behavior vector or vertex in to either a Strategy or DeterministicStrategy. If converting into a DeterministicStrategy, the vertex must contain Int64 values. Valid representations are "non-signaling", "normalized", and "generalized".

source
Base.convert — Method

DeterministicStrategy -> Vertex (Vector{Int64})

convert(
+    ::Type{Vector{Int64}},
+    strategy :: DeterministicStrategy;
+    rep = "normalized" :: String
+)
source
diff --git a/v0.1.3/LocalPolytope/adjacency_decomposition/index.html b/v0.1.3/LocalPolytope/adjacency_decomposition/index.html new file mode 100644 index 0000000..db0de07 --- /dev/null +++ b/v0.1.3/LocalPolytope/adjacency_decomposition/index.html @@ -0,0 +1,16 @@ + +Adjacency Decomposition · BellScenario.jl

Adjacency Decomposition

BellScenario.LocalPolytope.adjacency_decomposition — Function
adjacenecy_decomposition(
+    vertices :: Vector{Vector{Int64}},
+    BG_seed :: BellGame,
+    scenario :: LocalSignaling;
+    kwargs...
+) :: Dict

Given a polytpe represented by vertices, returns the complete set of canonical facets for prepare and measure scenario scenario. The adjacencydecomposition algorithm requires a seeded vertex which is supplied with the `BGseed` argument. Facets are returned in the lexicographic normal form.

Returned Dictionary Format

Returns a dictionary where the keys are canonical BellGames and the value is a dictionary with keys

  • "considered" => true, if the facet was considered in the algorithm.
  • "skipped" => true, if the facet was skipped (not considered).
  • "num_vertices" => number of vertices.
  • "norm_facet" => facet vector representative (normalized rep) of canonical game

Keyword Arguments: kwargs...

  • skip_games = [] :: Vector{BellGame} - List of games to skip.
  • max_vertices = 100 :: Int64 - The maximum number of vertices to allow in target facets.
  • dir = "./" :: String- Directory in which to createporta_tmp/and.json` files.
  • log = false :: Bool - If true, the facet dictionary is written to .json each iteration.
  • log_filename = "adjacency_decomposition_now.json" :: String
source
BellScenario.LocalPolytope.adjacent_facets — Function
adjacent_facets(
+    vertices :: Vector{Vector{Int64}},
+    F :: Vector{int64};
+    dir = "./" :: String,
+    cleanup = true ::Bool
+) :: Vector{Vector{Int64}}

For the polytope represented by vertices, returns the set of facets adjacent to F.

Facet vector F and the return facet vectors are assumed to ba in the normalized subspace.

The dir argument specifies where to where to write files and directories from XPORTA.jl. If cleanup is true, then a porta_tmp directory is created as a subdirectory of dir.

If cleanup is false, the created porta_tmp directory is not removed.

source
BellScenario.LocalPolytope.rotate_facet — Function
rotate_facet(
+    F :: Vector{Int64},
+    G :: Vector{Int64},
+    xbar :: Vector{Int64}
+) :: Vector{Int64}

Performs a rotation of facet F relative to non-included vertex xbar and returns the rotated facet vector. F is a polytope facet, G is a subfacet of F and xbar is a vertex not contained by F. By construction, the rotated facet contains xbar.

source
diff --git a/v0.1.3/LocalPolytope/facets/index.html b/v0.1.3/LocalPolytope/facets/index.html new file mode 100644 index 0000000..3e38a67 --- /dev/null +++ b/v0.1.3/LocalPolytope/facets/index.html @@ -0,0 +1,17 @@ + +Facets · BellScenario.jl

Facet Enumeration

BellScenario.LocalPolytope.facets — Function
facets(
+    vertices :: Vector{Vector{Int64}};
+    dir = "./" :: String,
+    cleanup=true :: Bool
+) :: Dict{String, Vector{Vector{Int64}}}

Computes the facet inequalities and equalities which bound the convex polyhedron defined by the set of vertices. If the optimal representation is used for the vertices, then no equalitities will be present. For example, if the "generalized" vertex representation is used the normalization constraints will be included in the resulting equalities. The output of this method is structured:

Dict(
+    "facets" => inequalities, # :: Vector{Vector{Int64}}
+    "equalitites" => equalities, # :: Vector{Vector{Int64}}
+)

Facet inequalities and equalities are represented as single vector f where f[1:(end-1)] contains the coefficients of the linear inquality and f[end] is the bound. Facet inequalities and equalities act upon a vertex v where inequalities are arranged such that there is an implicit f[1:(end-1)]' * v ≤ f[end] and equalities are arranged such that f[1:(end-1)]' * v == f[end]

Supporting Software

The vertex -> facet transformation is performed using the traf method of XPORTA.jl. Please refer to the source code for more details.

Performance Limitations

Vertex -> facet transformations are notoriously difficult computational problems. The performance limits of this method will be met far before the limits the vertex enumeration methods.

source
facets(poly::Polyhedron) :: Vector{Vector{Int64}}

Returns the facets of the local polytope poly. If the facets have not already been computed, then they are transformed into the half-space representation from the vertex representation.

source

Linear Programming of Facets

BellScenario.LocalPolytope.linear_nonclassicality_witness — Function
linear_nonclassicality_witness(
+    vertices :: Vector{Vector{T}} where T <: Real,
+    test_behavior :: Vector{<:Real};
+    verbose=false :: Bool
+) :: Vector{Float64}

Obtains a linear inequality $(\mathbf{G}^\star, \beta^\star)$ bounding the convex hull of the set of vertices ($\mathcal{V}$) and is violated by the test_behavior $(\mathbf{P}\notin \text{Conv}(\mathcal{V}))$. This task is achieved using the linear program described by Brunner et al. in Eq. 19 of https://arxiv.org/abs/1303.2849. The linear program is

\[\begin{align} + \min_{(\mathbf{G}, \beta)} \quad & \langle \mathbf{G},\mathbf{P}\rangle - \beta & \notag\\ + \text{s.t.} \quad & \langle \mathbf{G}, \mathbf{V} \rangle - \beta \leq 0 & \quad \forall \;\; \mathbf{V} \in \mathcal{V} \notag\\ + & \langle \mathbf{G}, \mathbf{P} \rangle \leq 1 & \notag\\ +\end{align}\]

The solution to the linear program is a linear nonclassicality witness $(\mathbf{G}^\star, \beta^\star)$ becuase the constraint $\lange\mathbf{G}^\star, \mathbf{V} \rangle - \beta^\star \leq 0$ ensures that no behavior in the polytope $\text{Conv}(\mathcal{V})$ can violate the inequality. Provided that $\mathbf{P} \notin \text{Conv}(\mathcal{V})$ technique the output linear inequality witnesses the nonclassicality of the test_behavior.

The optimized facet inequality $(\mathbf{G}^\star, \beta^\star)$ is returned as a vector $(G^\star_{0,0}, \dots, G^\star_{Y,X}, -\beta^\star)$ where $G^\star_{y,x}$ are the elements of $\mathbf{G}^\star$.

Supporting Software

The linear programming is performed using HiGHS solver via the JuMP interface. Please refer to the source code for more details.

Converting Output into Bell Game

The linear programming software outputs numerical values that have numerical error. Moreover, the linear inequality is scaled such that the classical bound is zero and the test_behavior score is one. In order to convert the output inequality into a BellGame, care must be taken to obtain the correct scaling factor to ensure that elements are integers.

Classical Test Behavior

If the test_behavior $\mathbf{P}$ is classical, meaning it satisfies $\mathbf{P}\in\text{Conv}(\mathcal{V})$, then the zero vector is returned as the optimal solution. Note that if all elements of $\mathbf{G}^\star$ satisfy $G^\star_{y,x}=0$, then all behaviors $\mathbf{P} \in \text{Conv}(\mathcal{V})$ are trivially optimal as $\langle\mathbf{G}^{\star}, \mathbf{P} \rangle - \beta^\star \leq 0$.

source
diff --git a/v0.1.3/LocalPolytope/generators/index.html b/v0.1.3/LocalPolytope/generators/index.html new file mode 100644 index 0000000..3ce2f38 --- /dev/null +++ b/v0.1.3/LocalPolytope/generators/index.html @@ -0,0 +1,5 @@ + +Generators · BellScenario.jl

Generating Vertices and Facets

BellScenario.LocalPolytope.generator_facet — Function
generator_facet( BG :: BellGame, scenario :: LocalSignaling ) :: BellGame

Finds the generating facet for the provided BellGame. The generator is provided in lexicographic normal form. The generating facet is found recursively by an algorithm which sorts by lexicographic scores.

source
diff --git a/v0.1.3/LocalPolytope/overview/index.html b/v0.1.3/LocalPolytope/overview/index.html new file mode 100644 index 0000000..8c3614d --- /dev/null +++ b/v0.1.3/LocalPolytope/overview/index.html @@ -0,0 +1,2 @@ + +Overview · BellScenario.jl

Local Polytope Overview

BellScenario.LocalPolytope — Module

Compute the bounds of classical Bell scenarios.

The LocalPolytope.jl module provides tools for characterizing the convex polyhedral structure of the correlations attainable using classical resources in a Bell scenario.

Each distinct strategy for a given Bell scenario corresponds to a point in vector space. The complete set of attainable strategies for a particular Bell scenario is denoted $\mathbf{P}$. Given the constraints of normalization and non-negativity, the extreme points of $\mathbf{P}$ correspond to deterministic strategies (matrices with 0,1 elements). Furthermore, when shared randomness is used in a Bell scenario, any two strategies can be mixed together in a convex combination. Hence, $\mathbf{P}$ is a convex polyhedron referred to as the local polytope.

A convex polytope has two equivalent descriptions:

  1. V-Description: The polytope is the convex hull of a set of extreme points known as vertices $\mathbf{V}$, $\quad\mathbf{P} = \text{conv}(\mathbf{V})$.
  2. H-Description: The polytope is the intersection of a set of linear half-spaces known as facets $\mathbf{F}$, $\quad\mathbf{P} = \cap\mathbf{F}$.

These two descriptions are equivalent,

\[\mathbf{P} = \text{conv}(\mathbf{V}) = \cap\mathbf{F},\]

and there exists a transformation between the set of vertices and facets $\mathbf{V} \leftrightarrow \mathbf{F}$.

For a given Bell scenario, the V-Description is typically easier to compute, however, the H-Description describes the bounds of the local polytope as as linear inequalities known as Bell inequalities. Bell inequalities are important because they provide a simple test to verify that a strategy is not contained by the local polyotpe. That is, if a Bell inequality is violated, the classical resources considered for the local polytope are not sufficient to reproduce the strategy. Hence, a Bell violation witnesses the use of resources with greater operational value. Bell violations are often used to characterize the advantages of quantum resources, however, it is important to note that a Bell violation can simply mean that more classical resources were used than anticipated.

Module Exports:

  • vertices - Compute the set of extreme-points for the local polytope.
  • num_vertices - The number of vertices for the local polytope.
  • vrep - Construct a Polyhedron in the vertex representation.
  • facets - Compute the linear inequalities which bound the local polytope.
  • generator_vertex - Provide a canonical form for a vertex.
  • generator_facet - Provide a canonical form for a facet.
  • adjacency_decomposition - Efficiently compute the generator facets for the local polytope using the adjacency decomposition technique.
source
diff --git a/v0.1.3/LocalPolytope/utils/index.html b/v0.1.3/LocalPolytope/utils/index.html new file mode 100644 index 0000000..2b20327 --- /dev/null +++ b/v0.1.3/LocalPolytope/utils/index.html @@ -0,0 +1,4 @@ + +Utilities · BellScenario.jl

Utilities for Facets and Vertices

BellScenario.LocalPolytope.dimension — Function
dimension( vertices :: Vector{Vector{Int64}} ) :: Int64

For the provided vertices (points), finds the dimension of the affine space spanned by the vertices. This method computes the dimension of a polytope, facet, or collection of points.

This also accepts matrices and DeterministicStrategy types as arguments:

dimension( vertices :: Vector{Matrix{Int64}} ) :: Int64
+
+dimension( vertices :: Vector{DeterministicStrategy} ) :: Int64
source
diff --git a/v0.1.3/LocalPolytope/vertices/index.html b/v0.1.3/LocalPolytope/vertices/index.html new file mode 100644 index 0000000..fd71a6e --- /dev/null +++ b/v0.1.3/LocalPolytope/vertices/index.html @@ -0,0 +1,11 @@ + +Vertices · BellScenario.jl

Vertices

Vertices are extreme points of the local polytope. They correspond to deterministic strategies. The vertex representation of a convex polytope can be computed via the Polyhedra.jl interface.

Polyhedra.vrep — Function
vrep(scenario::Scenario; vertices_kwargs...) :: XPORTA.Polyhedron

Constructs a Polyhedron using the vertex representation. See Polyhedra.jl for more details. The vertices_kwargs keyword arguments are passed through to the vertices function for each Scenario.

Return Type

This function differs from the Polyhedra.jl implementation in that it returns a Polyhedron type rather than a V-Representation. This is done to reduce the number of steps required to construct a polyhedron.

source

Vertex Enumeration

The vertices of each local polytope can be enumerated for the specified Bell Scenario.

BellScenario.LocalPolytope.vertices — Function
vertices( scenario :: BlackBox;
+    rep = "normalized" :: String
+) :: Vector{Vector{Int64}}

Generates the Local Polytope vertices for a BlackBox scenario. Valid represenations are:

*rep == "normalized" or rep == "generalized".

source
vertices( scenario :: LocalSignaling;
+    rep = "normalized" :: String
+    rank_d_only = false :: Bool
+) ::  Vector{Vector{Int64}}

Generates the deterministic strategies for the local polytope of LocalSignaling scenarios. The rank_d_only keyword arg specifies whether to exclude vertices which use fewer dits of communication and thus have a matrix rank less than d.

Warning

The vertices computed in this method are vectorized directly from a strategy matrix by column-majorization. These vertices are distinct from those produced by older LocalPolytope.vertices() methods which are row-majorized.

source
vertices( scenario :: BipartiteNonSignaling,
+    rep="non-signaling" :: String
+) :: Vector{Vector{Int64}}

Enumerates the LocalPolytope vertices for the BipartiteNonSignaling scenario. Valid representations for the vertices include:

  • "non-signaling", "normalized", "generalized"

A DomainError is thrown if a valid representation is not specified.

source

Vertex Counting

Count the number of local polytope vertices for the specified Bell Scenario.

BellScenario.LocalPolytope.num_vertices — Function
num_vertices( scenario :: BlackBox ) :: Int64

For $n$ outputs and $m$ inputs the number of vertices $|\mathcal{V}|$ are counted:

\[|\mathbf{V}| = n^m\]

source
num_vertices( scenario :: LocalSignaling;
+    rank_d_only = false :: Bool
+) :: Int64

If rank_d_only = true, then only strategies using d-dits are counted. For $X$ inputs and $Y$ outputs the number of vertices $|\mathcal{V}|$ are counted:

\[|\mathbf{V}| = \sum_{c=1}^d \left\{X \atop c \right\}\binom{Y}{c}c!\]

source
num_vertices( scenario :: BipartiteNonSignaling ) :: Int64

For two non-signaling black-boxes with $X$ and $Y$ inputs and $A$ and $B$ outputs respectively, the number of vertices $|\mathcal{V}|$ are counted:

\[|\mathbf{V}| = A^X B^Y\]

source

Vertex Dimension

Get the dimension of specified vertex representation.

BellScenario.LocalPolytope.vertex_dims — Function

For the given Scenario, returns the length of the vertex in the representation specified by rep. A DomainError is thrown if the rep is invalid.

vertex_dims(scenario :: Union{BlackBox,LocalSignaling}, rep :: String) :: Int64

Valid values of rep are "normalized" and "generalized".

source
vertex_dims( scenario:: BipartiteNonSignaling, rep :: String ) :: Int64

Valid values for rep include:

  • "non-signaling"
  • "normalized"
  • "generalized"
source
diff --git a/v0.1.3/Nonlocality/optimize_measurements/index.html b/v0.1.3/Nonlocality/optimize_measurements/index.html new file mode 100644 index 0000000..e5aba4a --- /dev/null +++ b/v0.1.3/Nonlocality/optimize_measurements/index.html @@ -0,0 +1,25 @@ + +Optimize Measurements · BellScenario.jl

Optimize Measurements

BellScenario.Nonlocality.optimize_measurement — Function

LocalSignaling scenario:

optimize_measurement(
+    scenario :: LocalSignaling,
+    game :: BellGame,
+    ρ_states :: Vector{<:AbstractMatrix}
+)

Finds the measurement that optimizes the score of the BellGame against the set of quantum states ρ_states. The optimization is performed with the following semi-definite program:

\[\begin{aligned} +&\max_{\{\Pi_y\}} \sum_{x,y} G_{x,y} \text{Tr}[\Pi_y \rho_x] \\ +&s.t. \quad \sum_y \Pi_y = \mathbb{I}, \quad \Pi_y \geq 0 +\end{aligned}\]

source

BipartiteNonSignaling scenario:

optimize_measurement(
+    game :: BellGame,
+    scenario :: BipartiteNonSignaling,
+    ρ_AB :: AbstractMatrix;
+    A_POVMs :: Vector{<:AbstractVector{<:AbstractMatrix}},
+) :: Dict

Find Bob's measurement which optimizes a BellGame's score for the shared quantum state ρ_AB and POVM measurement applied by Alice. The following semi-definite program optimizes the Bob's POVM:

\[\begin{aligned} + &\max_{\{\Pi_b^y\}} \sum_{a,b,x,y} G_{a,b,x,y}\text{Tr}[(\Pi_a^x \otimes \Pi_b^y)\rho_{AB}] \\ + &s.t. \quad \sum_b \Pi_b^y = \mathbb{I},\quad \Pi_b^y \geq 0 \quad \forall\; y +\end{aligned}\]

optimize_measurement(
+    game :: BellGame,
+    scenario :: BipartiteNonSignaling,
+    ρ_AB :: AbstractMatrix;
+    B_POVMs :: Vector{<:AbstractVector{<:AbstractMatrix}},
+) :: Dict

Find Alice's measurement which optimizes a BellGame's score for the shared quantum state ρ_{AB} and POVM measurement applied by Bob. The following semi-definite program optimizes the Alice's POVM:

\[\begin{aligned} +&\max_{\{\Pi_a^x\}} \sum_{a,b,x,y} G_{a,b,x,y}\text{Tr}[(\Pi_a^x \otimes \Pi_b^y)\rho_{AB}] \\ +&s.t. \quad \sum_a \Pi_a^x = \mathbb{I},\quad \Pi_a^x \geq 0 \quad \forall \;x +\end{aligned}\]

source
diff --git a/v0.1.3/Nonlocality/overview/index.html b/v0.1.3/Nonlocality/overview/index.html new file mode 100644 index 0000000..4671838 --- /dev/null +++ b/v0.1.3/Nonlocality/overview/index.html @@ -0,0 +1,2 @@ + +Overview · BellScenario.jl
diff --git a/v0.1.3/assets/custom.css b/v0.1.3/assets/custom.css new file mode 100644 index 0000000..a1216cd --- /dev/null +++ b/v0.1.3/assets/custom.css @@ -0,0 +1,10 @@ +#documenter nav.docs-sidebar a.docs-logo > img { + max-height: 12rem; +} + +img { + display: block; + width: 50%; + margin-left: auto; + margin-right: auto; +} diff --git a/v0.1.3/assets/documenter.js b/v0.1.3/assets/documenter.js new file mode 100644 index 0000000..f531160 --- /dev/null +++ b/v0.1.3/assets/documenter.js @@ -0,0 +1,889 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function () { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'minisearch'], function($, minisearch) { + +// In general, most search related things will have "search" as a prefix. +// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +let results = []; +let timer = undefined; + +let data = documenterSearchIndex["docs"].map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; +}); + +// list below is the lunr 2.1.3 list minus the intersect with names(Base) +// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) +// ideally we'd just filter the original list but it's not available as a variable +const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", +]); + +let index = new minisearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + boost: { title: 100 }, + fuzzy: 2, + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + tokenize: (string) => string.split(/[\s\-\.]+/), + }, +}); + +index.addAll(data); + +let filters = [...new Set(data.map((x) => x.category))]; +var modal_filters = make_modal_body_filters(filters); +var filter_results = []; + +$(document).on("keyup", ".documenter-search-input", function (event) { + // Adding a debounce to prevent disruptions from super-speed typing! + debounce(() => update_search(filter_results), 300); +}); + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + $(this).removeClass("search-filter-selected"); + } else { + $(this).addClass("search-filter-selected"); + } + + // Adding a debounce to prevent disruptions from crazy clicking! + debounce(() => get_filters(), 300); +}); + +/** + * A debounce function, takes a function and an optional timeout in milliseconds + * + * @function callback + * @param {number} timeout + */ +function debounce(callback, timeout = 300) { + clearTimeout(timer); + timer = setTimeout(callback, timeout); +} + +/** + * Make/Update the search component + * + * @param {string[]} selected_filters + */ +function update_search(selected_filters = []) { + let initial_search_body = ` +
Type something to get started!
+ `; + + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + results = index.search(querystring, { + filter: (result) => { + // Filtering results + if (selected_filters.length === 0) { + return result.score >= 1; + } else { + return ( + result.score >= 1 && selected_filters.includes(result.category) + ); + } + }, + }); + + let search_result_container = ``; + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + results.forEach(function (result) { + if (result.location) { + // Checking for duplication of results for the same page + if (!links.includes(result.location)) { + search_results += make_search_result(result, querystring); + count++; + } + + links.push(result.location); + } + }); + + let result_count = `
${count} result(s)
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + filter_results = []; + modal_filters = make_modal_body_filters(filters, filter_results); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(initial_search_body); + } +} + +/** + * Make the modal filter html + * + * @param {string[]} filters + * @param {string[]} selected_filters + * @returns string + */ +function make_modal_body_filters(filters, selected_filters = []) { + let str = ``; + + filters.forEach((val) => { + if (selected_filters.includes(val)) { + str += `${val}`; + } else { + str += `${val}`; + } + }); + + let filter_html = ` +
+ Filters: + ${str} +
+ `; + + return filter_html; +} + +/** + * Make the result component given a minisearch result data object and the value of the search input as queryString. + * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ +function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${result.title}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; +} + +/** + * Get selected filters, remake the filter html and lastly update the search modal + */ +function get_filters() { + let ele = $(".search-filters .search-filter-selected").get(); + filter_results = ele.map((x) => $(x).text().toLowerCase()); + modal_filters = make_modal_body_filters(filters, filter_results); + update_search(filter_results); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+`; + +let initial_search_body = ` +
Type something to get started!
+`; + +let search_modal_footer = ` + +`; + +$(document.body).append( + ` + + ` +); + +document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); +}); + +document.querySelector(".close-search-modal").addEventListener("click", () => { + closeModal(); +}); + +$(document).on("click", ".search-result-link", function () { + closeModal(); +}); + +document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; +}); + +// Functions to open and close a modal +function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); +} + +function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); +} + +document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.1.3/assets/logo.png b/v0.1.3/assets/logo.png new file mode 100644 index 0000000..b504127 Binary files /dev/null and b/v0.1.3/assets/logo.png differ diff --git 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files /dev/null and b/v0.1.3/assets/scenario_images/classical_chsh_scenario.png differ diff --git a/v0.1.3/assets/scenario_images/local_signaling_scenario.png b/v0.1.3/assets/scenario_images/local_signaling_scenario.png new file mode 100644 index 0000000..f3cda76 Binary files /dev/null and b/v0.1.3/assets/scenario_images/local_signaling_scenario.png differ diff --git a/v0.1.3/assets/themes/documenter-dark.css b/v0.1.3/assets/themes/documenter-dark.css new file mode 100644 index 0000000..9f5449f --- /dev/null +++ b/v0.1.3/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter 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form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark 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form.docs-search>input,fieldset[disabled] html.theme--documenter-dark .button,html.theme--documenter-dark fieldset[disabled] .button{cursor:not-allowed}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid rgba(0,0,0,0);border-radius:2px;border-right:0;border-top:0;content:" 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.button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter 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#documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--documenter-dark .image.is-3by4 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--documenter-dark .image.is-2by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--documenter-dark .image.is-2by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--documenter-dark .image.is-3by5 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--documenter-dark .image.is-3by5 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--documenter-dark .image.is-9by16 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--documenter-dark .image.is-9by16 .has-ratio,html.theme--documenter-dark #documenter 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.docs-logo>img.is-5by4{padding-top:80%}html.theme--documenter-dark .image.is-4by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--documenter-dark .image.is-3by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--documenter-dark .image.is-5by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--documenter-dark .image.is-16by9,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--documenter-dark .image.is-2by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--documenter-dark .image.is-3by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--documenter-dark .image.is-4by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--documenter-dark .image.is-3by4,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--documenter-dark .image.is-2by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--documenter-dark .image.is-3by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--documenter-dark .image.is-9by16,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--documenter-dark .image.is-1by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by2{padding-top:200%}html.theme--documenter-dark .image.is-1by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by3{padding-top:300%}html.theme--documenter-dark .image.is-16x16,html.theme--documenter-dark #documenter .docs-sidebar 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Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.1.3/assets/warner.js b/v0.1.3/assets/warner.js new file mode 100644 index 0000000..3f6f5d0 --- /dev/null +++ b/v0.1.3/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/v0.1.3/development_manual/index.html b/v0.1.3/development_manual/index.html new file mode 100644 index 0000000..936f960 --- /dev/null +++ b/v0.1.3/development_manual/index.html @@ -0,0 +1,2 @@ + +Development Manual · BellScenario.jl

Development Manual

Develop BellComm.jl Pkg

This project is packaged using Pkg.jl. For code changes to be reflected in the packaged software, the package must be set to development mode. This is done by running the command:

julia -e 'using Pkg; Pkg.develop("BellComm")'

Run Tests

Run the BellComm.jl package tests (continuous integration):

julia -e 'using Pkg;  Pkg.test("BellComm")'

Download test dependencies into local environment:

julia --project=test/ -e 'using Pkg; Pkg.instantiate()'

Run tests from a dev environment:

julia test/path/to/test_file.jl

Build Docs

Deploy docs server on local machine:

cd docs/build/; python3 -m http.server --bind localhost

Build docs from BellComm.jl source (continuous integration):

  1. Download docs dependencies into local environment:

     julia --project=docs/ -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()'

  1. Build docs:

     julia --project=docs/ docs/make.jl

Build docs from a dev environment:

julia -e 'using Pkg; Pkg.add("Documenter")'
julia docs/make.jl
diff --git a/v0.1.3/index.html b/v0.1.3/index.html new file mode 100644 index 0000000..db01b79 --- /dev/null +++ b/v0.1.3/index.html @@ -0,0 +1,2 @@ + +Home · BellScenario.jl

BellScenario.jl

Compute Bell inequalities and their quantum violations.

This software is a general tool for analyzing quantum non-locality in Bell scenarios.

Alpha Version

Breaking changes will be made in future commits.

  1. BellScenario: Types and constructors for describing Bell scenarios.
  2. LocalPolytope: Computes the bounds classical Bell Scenarios.
  3. Nonlocality: Optimizes quantum states and measurements for non-locality.

Overview

A Bell scenario describes a system of inter-connected black-box devices. A black-box is a simple device which takes an input and produces an output, however, no assumptions are made about the physical process inside the black-box. Black-boxes and Bell scenarios are completely characterized by their input-output statistics. The statistics of a Bell scenario depends on the underlying physics used by the black-box devices.

In most familiar cases, a black-box uses classical physics. Some examples of classical black-boxes include cars, slot machines, and software. Classical physics is deterministic, although, an observer of a classical black-box may not have access to the parameters required to predict the output. These parameters are called local hidden variables where local means that events occurring in one location of space-time do not affect another location. Essentially, classical black-boxes are deterministic while their local hidden variables create the perception of probabilistic behavior. Hence, any probabilistic mixture of deterministic black-box behaviors can, in principle, be realized for a given Bell scenario. The bounds of a classical Bell scenario's statistics are regarded as Bell inequalities and the set of all possible behaviors forms a convex polyhedron known as the local polytope.

Bell scenarios are of interest to fundamental physics because not all physical systems adhere to the constraint of locality. For example, quantum Bell scenarios are able to realize statistics unattainable by their corresponding classical systems. This non-local behavior manifests as a strong correlation between distant points in space-time. The non-local correlations of quantum systems result in operational advantages in communications, information security, and data processing applications.

While the success of many future technologies depends on realizing quantum non-locality, this task is very difficult, even in a theoretical sense. To witness non-locality, the local bounds must first be derived. Then, the parameters of the equivalent quantum system must be tuned to violate the derived Bell inequalities. Each of these problems are challenging analytically and numerically. The BellScenario.jl package provides the basic toolkit for finding non-locality in quantum systems.

Contents

Acknowledgements

Development of BellScenario.jl was made possible by the advisory of Dr. Eric Chitambar and general support from the Physics Department at the University of Illinois Urbana-Champaign. Funding was provided by NSF Award 1914440.

Citing

To reference this work, see CITATION.bib.

Licensing

BellScenario.jl is released under the MIT License.

diff --git a/v0.1.3/porta.log b/v0.1.3/porta.log new file mode 100644 index 0000000..48ca8e1 --- /dev/null +++ b/v0.1.3/porta.log @@ -0,0 +1,49 @@ + + + +log for /home/runner/.julia/artifacts/c4069199178afeecd74780d7e158664e253b3eea/bin/xporta -T ./porta_tmp/traf_tmp.poi + +input file ./porta_tmp/traf_tmp.poi o.k. +dimension : 8 +number of cone-points : 0 +number of conv-points : 16 + +GAUSS - ELIMINATION: +| iter- | # equa | # ineq | max| long| non- | mem | cpu-time | real-time | +| ation | | | bit-|arith| zeros | used | used | used | +| | | |length|metic| in % | in kB | in sec | in sec | +|-------|------------|----------|------|-----|--------|---------|-----------|-----------| +| 9 | 9 | 1 | 1 | n | 0.37 | 50 | 0.00 | 0.00 | +| 8 | 8 | 2 | 1 | n | 0.34 | 50 | 0.00 | 0.00 | +| 7 | 7 | 3 | 2 | n | 0.40 | 50 | 0.00 | 0.00 | +| 6 | 6 | 4 | 2 | n | 0.47 | 50 | 0.00 | 0.00 | +| 5 | 5 | 5 | 2 | n | 0.42 | 50 | 0.00 | 0.00 | +| 4 | 4 | 6 | 2 | n | 0.43 | 50 | 0.00 | 0.00 | +| 3 | 3 | 7 | 1 | n | 0.43 | 50 | 0.00 | 0.00 | +| 2 | 2 | 8 | 1 | n | 0.49 | 50 | 0.00 | 0.00 | +| 1 | 1 | 9 | 1 | n | 0.53 | 50 | 0.00 | 0.00 | + +FOURIER - MOTZKIN - ELIMINATION: +| iter- | upper | # ineq | max| long| non- | mem | cpu-time | real-time | +| ation | bound | | bit-|arith| zeros | used | used | used | +| | # ineq | |length|metic| in % | in kB | in sec | in sec | +|-------|------------|----------|------|-----|--------|---------|-----------|-----------| +| 7 | 10 | 10 | 1 | n | 0.12 | 846 | 0.00 | 0.00 | +| 6 | 11 | 11 | 1 | n | 0.14 | 846 | 0.00 | 0.00 | +| 5 | 12 | 12 | 1 | n | 0.15 | 846 | 0.00 | 0.00 | +| 4 | 18 | 18 | 1 | n | 0.26 | 847 | 0.00 | 0.00 | +| 3 | 33 | 21 | 1 | n | 0.27 | 847 | 0.00 | 0.00 | +| 2 | 36 | 24 | 1 | n | 0.28 | 847 | 0.00 | 0.00 | +| 1 | 52 | 24 | 1 | n | 0.30 | 847 | 0.00 | 0.00 | + +sum of inequalities over all iterations : 120 +maximal number of inequalities : 24 + +transformation to integer values +sorting system + +number of equations : 0 +number of inequalities : 24 + +output written to file ./porta_tmp/traf_tmp.poi.ieq + diff --git a/v0.1.3/search_index.js b/v0.1.3/search_index.js new file mode 100644 index 0000000..d62d4d8 --- /dev/null +++ b/v0.1.3/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"BellScenario/combinatorics/","page":"Combinatorics","title":"Combinatorics","text":"CurrentModule = BellScenario","category":"page"},{"location":"BellScenario/combinatorics/#Combinatorics-Utilities","page":"Combinatorics","title":"Combinatorics Utilities","text":"","category":"section"},{"location":"BellScenario/combinatorics/#Set-Partitions","page":"Combinatorics","title":"Set Partitions","text":"","category":"section"},{"location":"BellScenario/combinatorics/","page":"Combinatorics","title":"Combinatorics","text":"stirling2\nstirling2_partitions\nstirling2_matrices","category":"page"},{"location":"BellScenario/combinatorics/#BellScenario.stirling2","page":"Combinatorics","title":"BellScenario.stirling2","text":"stirling2( n :: Int64, k :: Int64 ) :: Int64\n\nCounts the number of ways to partition n items into k unlabelled groups. This quantity is known as Stirling's number of the 2nd kind:\n\nleftn atop k right = frac1ksum_i=0^k (-1)^ibinomki(k-i)^n\n\nThrows a DomainError if inputs do not satisfy n ≥ k ≥ 1.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/combinatorics/#BellScenario.stirling2_partitions","page":"Combinatorics","title":"BellScenario.stirling2_partitions","text":"stirling2_partitions( n :: Int64, k :: Int64 ) :: Vector{Vector{Vector{Int64}}}\n\nEnumerates the unique partitions of n items into k unlabelled sets. Each partition is a vector containing a set of k vectors designating each group.\n\nE.g.\n\njulia> stirling2_partitions(4, 2)\n7-element Vector{Vector{Vector{Int64}}}:\n [[1, 2, 3], [4]]\n [[3], [1, 2, 4]]\n [[1, 2], [3, 4]]\n [[1, 3], [2, 4]]\n [[2], [1, 3, 4]]\n [[2, 3], [1, 4]]\n [[1], [2, 3, 4]]\n\nThis recursive algorithm was inspired by this blog.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/combinatorics/#BellScenario.stirling2_matrices","page":"Combinatorics","title":"BellScenario.stirling2_matrices","text":"stirling2_matrices( n :: Int64, k :: Int64 ) :: Vector{Matrix{Bool}}\n\nGenerates the set of matrices with k rows and n columns where rows correspond to the groups and columns are the grouped elements. A non-zero element designates that the column id is grouped into the corresponding row.\n\nE.g.\n\njulia> stirling2_matrices(4, 2)\n7-element Vector{Matrix{Bool}}:\n [1 1 1 0; 0 0 0 1]\n [0 0 1 0; 1 1 0 1]\n [1 1 0 0; 0 0 1 1]\n [1 0 1 0; 0 1 0 1]\n [0 1 0 0; 1 0 1 1]\n [0 1 1 0; 1 0 0 1]\n [1 0 0 0; 0 1 1 1]\n\nA DomainError is thrown if n ≥ k ≥ 1 is not satisfied.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/combinatorics/#Matrix-Constructions","page":"Combinatorics","title":"Matrix Constructions","text":"","category":"section"},{"location":"BellScenario/combinatorics/","page":"Combinatorics","title":"Combinatorics","text":"permutation_matrices\nn_choose_k_matrices","category":"page"},{"location":"BellScenario/combinatorics/#BellScenario.permutation_matrices","page":"Combinatorics","title":"BellScenario.permutation_matrices","text":"permutation_matrices( dim :: Int64 ) :: Vector{Matrix{Bool}}\n\nGenerates the set of square permutation matrices of dimension dim.\n\nE.g.\n\njulia> permutation_matrices(3)\n6-element Vector{Matrix{Bool}}:\n [1 0 0; 0 1 0; 0 0 1]\n [1 0 0; 0 0 1; 0 1 0]\n [0 1 0; 1 0 0; 0 0 1]\n [0 0 1; 1 0 0; 0 1 0]\n [0 1 0; 0 0 1; 1 0 0]\n [0 0 1; 0 1 0; 1 0 0]\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/combinatorics/#BellScenario.n_choose_k_matrices","page":"Combinatorics","title":"BellScenario.n_choose_k_matrices","text":"n_choose_k_matrices( n :: Int64, k :: Int64 ) :: Vector{Matrix{Bool}}\n\nGenerates a set of n by k matrices which represent all combinations of selecting k columns from n rows. Each column, contains a single non-zero element and k rows contain a non-zero element.\n\nE.g.\n\njulia> n_choose_k_matrices( 4, 2 )\n6-element Vector{Matrix{Bool}}:\n [1 0; 0 1; 0 0; 0 0]\n [1 0; 0 0; 0 1; 0 0]\n [1 0; 0 0; 0 0; 0 1]\n [0 0; 1 0; 0 1; 0 0]\n [0 0; 1 0; 0 0; 0 1]\n [0 0; 0 0; 1 0; 0 1]\n\nA DomainError is thrown if n ≥ k ≥ 1 is not satisfied.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/combinatorics/#Alternative-Number-Bases","page":"Combinatorics","title":"Alternative Number Bases","text":"","category":"section"},{"location":"BellScenario/combinatorics/","page":"Combinatorics","title":"Combinatorics","text":"base_n_val","category":"page"},{"location":"BellScenario/combinatorics/#BellScenario.base_n_val","page":"Combinatorics","title":"BellScenario.base_n_val","text":"base_n_val(\n num_array :: Vector{Int64},\n base :: Int64;\n big_endian=true :: Bool\n) :: Int64\n\nGiven an array representing a number in base-n returns the value of that number in base-10.\n\nInputs:\n\nnum_array - Vector containing semi-positive integers less than base.\nbase - The base-n number represented by num_array.\nbig_endian - true if most significant place is at index 1, else false.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/overview/#BellScenario.jl-Overview","page":"Overview","title":"BellScenario.jl - Overview","text":"","category":"section"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"BellScenario","category":"page"},{"location":"BellScenario/overview/#BellScenario","page":"Overview","title":"BellScenario","text":"Types and constructors that represent Bell scenarios, their statistics, and their bounds.\n\nThe BellScenario.jl module is the base library for Bell scenario analysis.\n\nFeatures:\n\nScenario: abstract type describing Bell scenarios.\nAbstractStrategy: abstract type for Bell scenario statistics.\nAbstractGame: abstract type for Bell scenario bounds.\n\nA game theoretic framework is used to evaluate the performance of black-box systems. A cost function testing the black-box system is regarded as a game while the statistics generated by the black-box system are regarded as a strategy. A strategy is played against a game to achieve a score, hence, this framework provides a quantitative metric of the performance of a black-box system in regards to a particular task.\n\n\n\n\n\n","category":"module"},{"location":"BellScenario/overview/#Black-Box-Devices","page":"Overview","title":"Black-Box Devices","text":"","category":"section"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"A black-box device describes a physical system with inputs and outputs that an observer can test, however, the observer has no knowledge of the physical system contained within the device.","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"(Image: Black-Box Device)","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"A black-box device accepts and input x and produces an output y. No assumptions are made about how y is computed from x. A black-box is characterized by its conditional probability distribution S(yx) which can be derived by collecting the input-output statistics of the black-box. The distribution S(yx) is a stochastic map taking input x to output y. In BellScenario.jl, stochastic maps are referred to as strategies and outlined in greater detail in the BellScenario.jl - Strategies section.","category":"page"},{"location":"BellScenario/overview/#Resources","page":"Overview","title":"Resources","text":"","category":"section"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"The statistics achieved by Bell scenario depend on the resources available in the black-box system. In the context of Bell scenarios, resources describe the connection between black-boxes. Resources are typically limited and place restrictions on the black-box system.","category":"page"},{"location":"BellScenario/overview/#Communication-Resources:","page":"Overview","title":"Communication Resources:","text":"","category":"section"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"Two (or more) black-boxes may coordinate using a limited amount of communication.","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"Classical Communication - A classical state is sent from one black-box to another.\nQuantum Communication - A quantum state is sent from one black-box to another.","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"Communication resources are used during a Bell test. Classical communication resources are measured in dits which describes the number of distinct messages that can be sent. Quantum communication resources are measured in qudits which describes the Hilbert space dimension of the quantum state used to communicate. Quantum communication can simulate classical communication by encoding the classical messages as an orthogonal basis on the qudit Hilbert space.","category":"page"},{"location":"BellScenario/overview/#Correlation-Resources:","page":"Overview","title":"Correlation Resources:","text":"","category":"section"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"Two (or more) black-boxes may coordinate using a limited amount of correlation.","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"Shared Randomness - Two or more black-boxes share a random variable lambda drawn from a sample space Lambda with probability q(lambda).\nQuantum Entanglement - Two or more black-boxes share a non-separable quantum state.","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"Correlation resources may be distributed before any Bell tests are run.","category":"page"},{"location":"BellScenario/overview/#Diagrams","page":"Overview","title":"Diagrams","text":"","category":"section"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"A Bell scenario is depicted by a Directed Acyclic Graph (DAG) which shows the causal flow of information through the black-box system. For example, the following figure shows a bipartite signaling scenario.","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"(Image: Bipartite Signaling Scenario Diagram)","category":"page"},{"location":"BellScenario/overview/","page":"Overview","title":"Overview","text":"In the diagram, the information flows from inputs x and y to outputs a and b. In general, a free origin of an arrow corresponds to an input of the Bell scenario while a free point of an arrow corresponds to an output of the Bell scenario. The blue rectangular nodes in the graph represent black-box devices, e.g. A_lambda(ax) and B_lambda(bmy). When communication is present between two black-boxes, it is depicted by a solid arrow originating at one node and terminating at another. The red circular nodes in the graph, e.g. Lambda, represent correlation resources. When a correlation resource is shared between two black-boxes, it is depicted by a red dotted arrow flowing from the correlation resource node to the each of the black-box nodes.","category":"page"},{"location":"LocalPolytope/utils/","page":"Utilities","title":"Utilities","text":"CurrentModule = LocalPolytope","category":"page"},{"location":"LocalPolytope/utils/#Utilities-for-Facets-and-Vertices","page":"Utilities","title":"Utilities for Facets and Vertices","text":"","category":"section"},{"location":"LocalPolytope/utils/","page":"Utilities","title":"Utilities","text":"dimension","category":"page"},{"location":"LocalPolytope/utils/#BellScenario.LocalPolytope.dimension","page":"Utilities","title":"BellScenario.LocalPolytope.dimension","text":"dimension( vertices :: Vector{Vector{Int64}} ) :: Int64\n\nFor the provided vertices (points), finds the dimension of the affine space spanned by the vertices. This method computes the dimension of a polytope, facet, or collection of points.\n\nThis also accepts matrices and DeterministicStrategy types as arguments:\n\ndimension( vertices :: Vector{Matrix{Int64}} ) :: Int64\n\ndimension( vertices :: Vector{DeterministicStrategy} ) :: Int64\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"CurrentModule = BellScenario","category":"page"},{"location":"BellScenario/strategies/#BellScenario.jl-Strategies","page":"Strategies","title":"BellScenario.jl - Strategies","text":"","category":"section"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"AbstractStrategy\nStrategy\nstrategy_dims\nrandom_strategy","category":"page"},{"location":"BellScenario/strategies/#BellScenario.AbstractStrategy","page":"Strategies","title":"BellScenario.AbstractStrategy","text":"An AbstractStrategy is an abstract type parent to all strategy matrices. A black-box device is characterized by its conditional probabilities which can be organized into a strategy matrix or column-stochastic map S mathcalX to mathcalY. The strategy matrix S is constructed as follows,\n\nS = sum_xy P(yx) yranglelangle x\n\nwhere the elements of a strategy matrix must be non-negative and normalized:\n\nP(yx) geq 0\nsum_y P(yx) = 1\n\nSince strategies are just stochastic matrices, the product of two strategies yields a new strategy, e.g. S_A*S_B = S_C.\n\n*(S1::AbstractStrategy, S2::AbstractStrategy) :: Strategy\n\nWhen multiplied together, two strategies may represent a new Scenario, hence, a the multiplication operator can also be passed a Scenario.\n\n*(S1::AbstractStrategy, S2::AbstractStrategy, scenario::Scenario) :: Strategy\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/strategies/#BellScenario.Strategy","page":"Strategies","title":"BellScenario.Strategy","text":"Strategy(conditionals :: Matrix{<:Real}; atol::Float64 = 1e-7) <: AbstractMatrix{Float64}\n\nStrategy(conditionals :: Conditionals; atol::Float64 = 1e-7) <: AbstractMatrix{Float64}\n\nThe conditionals parameter is a column stochastic matrix. The conditionals can either be accepted in a raw Matrix{<:Real} format or as a Conditionals type from QBase.jl. The atol parameter expresses the absolute tolerance for numerical error in the constraints of conditional probablities. By default, the constructor creates a strategy for a 'BlackBox' scenario. However, a Scenario can be passed to the Strategy constructor.\n\nStrategy(conditionals :: Matrix{<:Real}, scenario :: Scenario, atol::Float64=1e-7)\n\nErrors:\n\nA DomainError is thrown if:\n\nThe provided Scenario does not match the expected dimension of the conditionals matrix.\nThe conditionals matrix is not a valid stochastic matrix, e.g. non-negative and normalized.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/strategies/#BellScenario.strategy_dims","page":"Strategies","title":"BellScenario.strategy_dims","text":"strategy_dims( scenario :: Scenario ) :: Tuple{Int64, Int64}\n\nReturns the dimensions of the Conditionals matrix describing a Strategy for the Scenario at hand. Each Scenario, can place unique constraints on the matrix dimensions, therfore, separate methods are called for each concrete Scenario.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/strategies/#BellScenario.random_strategy","page":"Strategies","title":"BellScenario.random_strategy","text":"random_strategy(\n num_inputs :: Int64,\n num_outputs :: Int64;\n) :: Strategy\n\nConstructs a randomized Strategy matrix. Zeros are inserted into the strategy to ensure that it does not closely resemble a uniform distribution.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/strategies/#Deterministic-Strategies","page":"Strategies","title":"Deterministic Strategies","text":"","category":"section"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"DeterministicStrategy\nis_deterministic\ndeterministic_strategies","category":"page"},{"location":"BellScenario/strategies/#BellScenario.DeterministicStrategy","page":"Strategies","title":"BellScenario.DeterministicStrategy","text":"DeterministicStrategy(conditionals :: Matrix{Int64}) <: AbstractStrategy{Int64}\n\nA strategy matrix describing the behavior of a deterministic black-box. A strategy deterministic if its elements satisfy P(yx)in01 in addition to the non-negativity and normalization constraints. By default, the constructor creates a strategy for a 'BlackBox' scenario, however, a Scenario can be passed to the DeterministicStrategy constructor.\n\nDeterministicStrategy(conditionals :: Matrix{Int}, scenario :: Scenario)\n\nThe product of two deterministic strategies is a DeterministicStrategy.\n\n*(S1::DeterministicStrategy, S2::DeterministicStrategy) :: DeterministicStrategy\n\nWhen multiplied a new Scenario can be specified.\n\n*(\n S1::DeterministicStrategy,\n S2::DeterministicStrategy,\n scenario::Scenario\n) :: Deterministic Strategy\n\nErrors:\n\nA DomainError is thrown if:\n\nThe provided Scenario does not match the dimension of the conditionals matrix.\nThe elements of conditionals are not 0 or 1.\nThe strategy elements are not non-negative and normalized.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/strategies/#BellScenario.is_deterministic","page":"Strategies","title":"BellScenario.is_deterministic","text":"is_deterministic( strategy :: AbstractMatrix ) :: Bool\n\nReturns true if all elements of strategy are either 0 or 1 and the matrix is a valid conditional probability distribution.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/strategies/#BellScenario.deterministic_strategies","page":"Strategies","title":"BellScenario.deterministic_strategies","text":"deterministic_strategies(scenario :: BlackBox) :: Vector{Matrix{Int64}}\n\ndeterministic_strategies(num_out :: Int64, num_in :: Int64) :: Vector{Matrix{Int64}}\n\nEnumerates the set of deterministic strategies for the specified BlackBox. For performance, enumerated deterministic strategies are left as matrices and not constructed into DeterministicStrategy types.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/strategies/#Quantum-Strategies","page":"Strategies","title":"Quantum Strategies","text":"","category":"section"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"quantum_strategy","category":"page"},{"location":"BellScenario/strategies/#BellScenario.quantum_strategy","page":"Strategies","title":"BellScenario.quantum_strategy","text":"Constructs a strategy matrix given quantum states and measurements. The supported scenarios include:\n\nBlackBox scenarios:\n\nquantum_strategy(\n Π :: AbstractVector{<:AbstractMatrix},\n ρ_states :: Vector{<:State};\n atol=1e-7::Float64\n) :: Strategy\n\nFor a quantum system the conditional proabilities are constructed as\n\n P(yx) = textTrPi_yrho_x\n\n\n\n\n\nLocalSignaling scenarios:\n\nquantum_strategy(\n Π :: AbstractVector{<:AbstractMatrix},\n ρ_states :: Vector{<:AbstractMatrix},\n scenario :: LocalSignaling;\n atol=1e-7::Float64\n) :: Strategy\n\nFor quantum systems the conditional probabilities are construct as\n\n P(yx) = textTrPi_yrho_x\n\nA DomainError is thrown if the provided states and measurements are not compatible with the specified scenario.\n\n\n\n\n\nBipartiteNonSignaling scenarios:\n\nquantum_strategy(\n ρ_AB :: AbstractMatrix,\n Π_Ax :: Vector{<:AbstractVector{<:AbstractMatrix}},\n Π_By :: Vector{<:AbstractVector{<:AbstractMatrix}},\n scenario :: BipartiteNonSignaling;\n atol::Float64 = 1e-7\n)\n\nConstructs a strategy matrix in the generalized representation for the quantum system with conditional probabilities.\n\n P(abxy) = textTr(Pi_a^xotimesPi_b^y)rho_AB\n\nA DomainError is thrown if\n\nThe length of each POVM does not match the scenarios number of outputs\nThe number of each party's POVMS doesn't match the the number of inputs.\n\n\n\n\n\n","category":"function"},{"location":"BellScenario/strategies/#Behaviors","page":"Strategies","title":"Behaviors","text":"","category":"section"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"The conditional probabilities of a Bell scenario can be represented by data structures other than a Strategy matrix. Most notably, the conditional probabilities can be organized into a vector referred to as a behavior in the literature. A behavior is a column-major vectorization of a strategy matrix e.g. S[:]. The dimension of the behavior vector (or strategy matrix) can be reduced to an equivalent subspace by using the normalization and non-signaling constraints to removed redundant parameters of the conditional probability distribution.","category":"page"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"Subspaces used within BellScenario.jl include:","category":"page"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"\"generalized\" - All conditional probabilities P(yx) for the Bell scenario are present.\n\"normalized\" - The normalization constraint on each column is used to remove the last row of the strategy matrix.\n\"non-signaling\" - The non-signaling constraints are applied to reduce the \"normalized\" subspace further.","category":"page"},{"location":"BellScenario/strategies/#Conversion-Methods","page":"Strategies","title":"Conversion Methods","text":"","category":"section"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"Strategy matrices and behavior vectors are isomorphic representations. Therefore, a behavior can be converted into a strategy and vice versa.","category":"page"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"Currently, behaviors are loosely represented by Vector{Int64} and Vector{Float64} types and a rep argument is used to specify the subspace. This may change in future updates of BellScenario.jl.","category":"page"},{"location":"BellScenario/strategies/","page":"Strategies","title":"Strategies","text":"convert(::Type{DeterministicStrategy}, ::Vector{Int64}, ::BlackBox; ::String)\nconvert(::Type{<:AbstractStrategy}, ::Vector{Float64}, ::BipartiteNonSignaling; ::String)\nconvert(::Type{Vector{Int64}}, ::DeterministicStrategy; ::String)","category":"page"},{"location":"BellScenario/strategies/#Base.convert-Tuple{Type{DeterministicStrategy}, Vector{Int64}, BlackBox}","page":"Strategies","title":"Base.convert","text":"Vertex (Vector{Int64}) -> DeterministicStrategy\n\nconvert(\n ::Type{DeterministicStrategy},\n vertex :: Vector{Int64},\n scenario :: Scenario;\n rep = \"normalized\" :: String\n)\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/strategies/#Base.convert-Tuple{Type{<:AbstractStrategy}, Vector{Float64}, BipartiteNonSignaling}","page":"Strategies","title":"Base.convert","text":"convert(\n S :: Type{<:AbstractStrategy}, vertex::Vector{<:Real}, scenario::BipartiteNonSignaling;\n rep=\"non-signaling\" :: String\n)\n\nTransforms a behavior vector or vertex in to either a Strategy or DeterministicStrategy. If converting into a DeterministicStrategy, the vertex must contain Int64 values. Valid representations are \"non-signaling\", \"normalized\", and \"generalized\".\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/strategies/#Base.convert-Tuple{Type{Vector{Int64}}, DeterministicStrategy}","page":"Strategies","title":"Base.convert","text":"DeterministicStrategy -> Vertex (Vector{Int64})\n\nconvert(\n ::Type{Vector{Int64}},\n strategy :: DeterministicStrategy;\n rep = \"normalized\" :: String\n)\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/scenarios/","page":"Scenarios","title":"Scenarios","text":"CurrentModule = BellScenario","category":"page"},{"location":"BellScenario/scenarios/#BellScenario.jl-Scenarios","page":"Scenarios","title":"BellScenario.jl - Scenarios","text":"","category":"section"},{"location":"BellScenario/scenarios/","page":"Scenarios","title":"Scenarios","text":"Scenario\nBlackBox\nLocalSignaling\nBipartiteNonSignaling\nBipartiteSignaling","category":"page"},{"location":"BellScenario/scenarios/#BellScenario.Scenario","page":"Scenarios","title":"BellScenario.Scenario","text":"A Scenario is an abstract type parent to all Bell scenarios. Each child of this abstract type describes a distinct black-box system configuration.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/scenarios/#BellScenario.BlackBox","page":"Scenarios","title":"BellScenario.BlackBox","text":"BlackBox(num_out :: Int64, num_in :: Int64) <: Scenario\n\nA Bell scenario consisting of a single black-box device with num_in inputs and num_out outputs.\n\n(Image: Black-Box Device)\n\nThe black-box device computes the output y from the input x by performing a stochastic map S(yx).\n\nErrors\n\nA DomainError is thrown if parameters num_out or num_in is less than 1.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/scenarios/#BellScenario.LocalSignaling","page":"Scenarios","title":"BellScenario.LocalSignaling","text":"LocalSignaling(\n X :: Int64,\n Y :: Int64,\n d :: Int64,\n) <: Scenario\n\nA bipartite signaling scenario where information is passed from a transmitter black-box to a receiver black-box using no more than d dits of communication.\n\n(Image: Local Signaling Scenario)\n\nThe transmitter device has X inputs and the receiver device has Y outputs and shared randomness is held between the two devices. When quantum communication is used instead of classical communication no Bell violations occur.\n\nErrors\n\nA DomainError is thrown if X, Y, or d is less than 1.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/scenarios/#BellScenario.BipartiteNonSignaling","page":"Scenarios","title":"BellScenario.BipartiteNonSignaling","text":"BipartiteNonSignaling(\n A :: Int64,\n B :: Int64,\n X :: Int64,\n Y :: Int64\n) <: Scenario\n\nA bipartite non-signaling scenario where each device receives an input and produces an output. Let Alice be the device with A outputs and X inputs while Bob is the device with B outputs and Y inputs.\n\n(Image: Bipartite Non-Signaling Scenario)\n\nShared randomness is held between Alice and Bob. When Alice and Bob share quantum entanglement, Bell violations are known to occur.\n\nErrors\n\nA DomainError is thrown if A, B, X, or Y is less than 1.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/scenarios/#BellScenario.BipartiteSignaling","page":"Scenarios","title":"BellScenario.BipartiteSignaling","text":"BipartiteSignaling(\n A :: Tuple{Int64, Int64},\n B :: Tuple{Int64, Int64};\n dits :: Int64 = 1,\n bidirectional :: Bool = false\n) <: Scenario\n\nA bipartite signaling scenario where each device can send a message to the other.\n\n(Image: Bipartite Signaling Scenario)\n\n\n\n\n\n","category":"type"},{"location":"LocalPolytope/generators/","page":"Generators","title":"Generators","text":"CurrentModule = LocalPolytope","category":"page"},{"location":"LocalPolytope/generators/#Generating-Vertices-and-Facets","page":"Generators","title":"Generating Vertices and Facets","text":"","category":"section"},{"location":"LocalPolytope/generators/","page":"Generators","title":"Generators","text":"generator_vertex\ngenerator_facet","category":"page"},{"location":"LocalPolytope/generators/#BellScenario.LocalPolytope.generator_vertex","page":"Generators","title":"BellScenario.LocalPolytope.generator_vertex","text":"generator_vertex(\n D :: DeterministicStrategy,\n scenario :: LocalSignaling\n) :: DeterministicStrategy\n\nFinds the generating vertex for the provided DeterministicStrategy. The generating vertex is the lexicographic normal form of D.\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/generators/#BellScenario.LocalPolytope.generator_facet","page":"Generators","title":"BellScenario.LocalPolytope.generator_facet","text":"generator_facet( BG :: BellGame, scenario :: LocalSignaling ) :: BellGame\n\nFinds the generating facet for the provided BellGame. The generator is provided in lexicographic normal form. The generating facet is found recursively by an algorithm which sorts by lexicographic scores.\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"CurrentModule = LocalPolytope","category":"page"},{"location":"LocalPolytope/vertices/#Vertices","page":"Vertices","title":"Vertices","text":"","category":"section"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"Vertices are extreme points of the local polytope. They correspond to deterministic strategies. The vertex representation of a convex polytope can be computed via the Polyhedra.jl interface.","category":"page"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"vrep","category":"page"},{"location":"LocalPolytope/vertices/#Polyhedra.vrep","page":"Vertices","title":"Polyhedra.vrep","text":"vrep(scenario::Scenario; vertices_kwargs...) :: XPORTA.Polyhedron\n\nConstructs a Polyhedron using the vertex representation. See Polyhedra.jl for more details. The vertices_kwargs keyword arguments are passed through to the vertices function for each Scenario.\n\nnote: Return Type\nThis function differs from the Polyhedra.jl implementation in that it returns a Polyhedron type rather than a V-Representation. This is done to reduce the number of steps required to construct a polyhedron.\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/vertices/#Vertex-Enumeration","page":"Vertices","title":"Vertex Enumeration","text":"","category":"section"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"The vertices of each local polytope can be enumerated for the specified Bell Scenario.","category":"page"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"vertices","category":"page"},{"location":"LocalPolytope/vertices/#BellScenario.LocalPolytope.vertices","page":"Vertices","title":"BellScenario.LocalPolytope.vertices","text":"vertices( scenario :: BlackBox;\n rep = \"normalized\" :: String\n) :: Vector{Vector{Int64}}\n\nGenerates the Local Polytope vertices for a BlackBox scenario. Valid represenations are:\n\n*rep == \"normalized\" or rep == \"generalized\".\n\n\n\n\n\nvertices( scenario :: LocalSignaling;\n rep = \"normalized\" :: String\n rank_d_only = false :: Bool\n) :: Vector{Vector{Int64}}\n\nGenerates the deterministic strategies for the local polytope of LocalSignaling scenarios. The rank_d_only keyword arg specifies whether to exclude vertices which use fewer dits of communication and thus have a matrix rank less than d.\n\nwarning: Warning\nThe vertices computed in this method are vectorized directly from a strategy matrix by column-majorization. These vertices are distinct from those produced by older LocalPolytope.vertices() methods which are row-majorized.\n\n\n\n\n\nvertices( scenario :: BipartiteNonSignaling,\n rep=\"non-signaling\" :: String\n) :: Vector{Vector{Int64}}\n\nEnumerates the LocalPolytope vertices for the BipartiteNonSignaling scenario. Valid representations for the vertices include:\n\n\"non-signaling\", \"normalized\", \"generalized\"\n\nA DomainError is thrown if a valid representation is not specified.\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/vertices/#Vertex-Counting","page":"Vertices","title":"Vertex Counting","text":"","category":"section"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"Count the number of local polytope vertices for the specified Bell Scenario.","category":"page"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"num_vertices","category":"page"},{"location":"LocalPolytope/vertices/#BellScenario.LocalPolytope.num_vertices","page":"Vertices","title":"BellScenario.LocalPolytope.num_vertices","text":"num_vertices( scenario :: BlackBox ) :: Int64\n\nFor n outputs and m inputs the number of vertices mathcalV are counted:\n\nmathbfV = n^m\n\n\n\n\n\nnum_vertices( scenario :: LocalSignaling;\n rank_d_only = false :: Bool\n) :: Int64\n\nIf rank_d_only = true, then only strategies using d-dits are counted. For X inputs and Y outputs the number of vertices mathcalV are counted:\n\nmathbfV = sum_c=1^d leftX atop c rightbinomYcc\n\n\n\n\n\nnum_vertices( scenario :: BipartiteNonSignaling ) :: Int64\n\nFor two non-signaling black-boxes with X and Y inputs and A and B outputs respectively, the number of vertices mathcalV are counted:\n\nmathbfV = A^X B^Y\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/vertices/#Vertex-Dimension","page":"Vertices","title":"Vertex Dimension","text":"","category":"section"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"Get the dimension of specified vertex representation.","category":"page"},{"location":"LocalPolytope/vertices/","page":"Vertices","title":"Vertices","text":"vertex_dims","category":"page"},{"location":"LocalPolytope/vertices/#BellScenario.LocalPolytope.vertex_dims","page":"Vertices","title":"BellScenario.LocalPolytope.vertex_dims","text":"For the given Scenario, returns the length of the vertex in the representation specified by rep. A DomainError is thrown if the rep is invalid.\n\nvertex_dims(scenario :: Union{BlackBox,LocalSignaling}, rep :: String) :: Int64\n\nValid values of rep are \"normalized\" and \"generalized\".\n\n\n\n\n\nvertex_dims( scenario:: BipartiteNonSignaling, rep :: String ) :: Int64\n\nValid values for rep include:\n\n\"non-signaling\"\n\"normalized\"\n\"generalized\"\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/facets/","page":"Facets","title":"Facets","text":"CurrentModule = LocalPolytope","category":"page"},{"location":"LocalPolytope/facets/#Facet-Enumeration","page":"Facets","title":"Facet Enumeration","text":"","category":"section"},{"location":"LocalPolytope/facets/","page":"Facets","title":"Facets","text":"facets","category":"page"},{"location":"LocalPolytope/facets/#BellScenario.LocalPolytope.facets","page":"Facets","title":"BellScenario.LocalPolytope.facets","text":"facets(\n vertices :: Vector{Vector{Int64}};\n dir = \"./\" :: String,\n cleanup=true :: Bool\n) :: Dict{String, Vector{Vector{Int64}}}\n\nComputes the facet inequalities and equalities which bound the convex polyhedron defined by the set of vertices. If the optimal representation is used for the vertices, then no equalitities will be present. For example, if the \"generalized\" vertex representation is used the normalization constraints will be included in the resulting equalities. The output of this method is structured:\n\nDict(\n \"facets\" => inequalities, # :: Vector{Vector{Int64}}\n \"equalitites\" => equalities, # :: Vector{Vector{Int64}}\n)\n\nFacet inequalities and equalities are represented as single vector f where f[1:(end-1)] contains the coefficients of the linear inquality and f[end] is the bound. Facet inequalities and equalities act upon a vertex v where inequalities are arranged such that there is an implicit f[1:(end-1)]' * v ≤ f[end] and equalities are arranged such that f[1:(end-1)]' * v == f[end]\n\nnote: Supporting Software\nThe vertex -> facet transformation is performed using the traf method of XPORTA.jl. Please refer to the source code for more details.\n\nwarning: Performance Limitations\nVertex -> facet transformations are notoriously difficult computational problems. The performance limits of this method will be met far before the limits the vertex enumeration methods.\n\n\n\n\n\nfacets(poly::Polyhedron) :: Vector{Vector{Int64}}\n\nReturns the facets of the local polytope poly. If the facets have not already been computed, then they are transformed into the half-space representation from the vertex representation.\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/facets/#Linear-Programming-of-Facets","page":"Facets","title":"Linear Programming of Facets","text":"","category":"section"},{"location":"LocalPolytope/facets/","page":"Facets","title":"Facets","text":"linear_nonclassicality_witness","category":"page"},{"location":"LocalPolytope/facets/#BellScenario.LocalPolytope.linear_nonclassicality_witness","page":"Facets","title":"BellScenario.LocalPolytope.linear_nonclassicality_witness","text":"linear_nonclassicality_witness(\n vertices :: Vector{Vector{T}} where T <: Real,\n test_behavior :: Vector{<:Real};\n verbose=false :: Bool\n) :: Vector{Float64}\n\nObtains a linear inequality (mathbfG^star beta^star) bounding the convex hull of the set of vertices (mathcalV) and is violated by the test_behavior (mathbfPnotin textConv(mathcalV)). This task is achieved using the linear program described by Brunner et al. in Eq. 19 of https://arxiv.org/abs/1303.2849. The linear program is\n\nbeginalign\n min_(mathbfG beta) quad langle mathbfGmathbfPrangle - beta notag\n textst quad langle mathbfG mathbfV rangle - beta leq 0 quad forall mathbfV in mathcalV notag\n langle mathbfG mathbfP rangle leq 1 notag\nendalign\n\nThe solution to the linear program is a linear nonclassicality witness (mathbfG^star beta^star) becuase the constraint langemathbfG^star mathbfV rangle - beta^star leq 0 ensures that no behavior in the polytope textConv(mathcalV) can violate the inequality. Provided that mathbfP notin textConv(mathcalV) technique the output linear inequality witnesses the nonclassicality of the test_behavior.\n\nThe optimized facet inequality (mathbfG^star beta^star) is returned as a vector (G^star_00 dots G^star_YX -beta^star) where G^star_yx are the elements of mathbfG^star.\n\nnote: Supporting Software\nThe linear programming is performed using HiGHS solver via the JuMP interface. Please refer to the source code for more details.\n\nnote: Converting Output into Bell Game\nThe linear programming software outputs numerical values that have numerical error. Moreover, the linear inequality is scaled such that the classical bound is zero and the test_behavior score is one. In order to convert the output inequality into a BellGame, care must be taken to obtain the correct scaling factor to ensure that elements are integers.\n\nnote: Classical Test Behavior\nIf the test_behavior mathbfP is classical, meaning it satisfies mathbfPintextConv(mathcalV), then the zero vector is returned as the optimal solution. Note that if all elements of mathbfG^star satisfy G^star_yx=0, then all behaviors mathbfP in textConv(mathcalV) are trivially optimal as langlemathbfG^star mathbfP rangle - beta^star leq 0.\n\n\n\n\n\n","category":"function"},{"location":"development_manual/#Development-Manual","page":"Development Manual","title":"Development Manual","text":"","category":"section"},{"location":"development_manual/#Develop-BellComm.jl-Pkg","page":"Development Manual","title":"Develop BellComm.jl Pkg","text":"","category":"section"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"This project is packaged using Pkg.jl. For code changes to be reflected in the packaged software, the package must be set to development mode. This is done by running the command:","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"julia -e 'using Pkg; Pkg.develop(\"BellComm\")'","category":"page"},{"location":"development_manual/#Run-Tests","page":"Development Manual","title":"Run Tests","text":"","category":"section"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Run the BellComm.jl package tests (continuous integration):","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"julia -e 'using Pkg; Pkg.test(\"BellComm\")'","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Download test dependencies into local environment:","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"julia --project=test/ -e 'using Pkg; Pkg.instantiate()'","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Run tests from a dev environment:","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"julia test/path/to/test_file.jl","category":"page"},{"location":"development_manual/#Build-Docs","page":"Development Manual","title":"Build Docs","text":"","category":"section"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Deploy docs server on local machine:","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"cd docs/build/; python3 -m http.server --bind localhost","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Build docs from BellComm.jl source (continuous integration):","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Download docs dependencies into local environment:\n julia --project=docs/ -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()'","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Build docs:\n julia --project=docs/ docs/make.jl","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"Build docs from a dev environment:","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"julia -e 'using Pkg; Pkg.add(\"Documenter\")'","category":"page"},{"location":"development_manual/","page":"Development Manual","title":"Development Manual","text":"julia docs/make.jl","category":"page"},{"location":"Nonlocality/optimize_measurements/","page":"Optimize Measurements","title":"Optimize Measurements","text":"CurrentModule = Nonlocality","category":"page"},{"location":"Nonlocality/optimize_measurements/#Optimize-Measurements","page":"Optimize Measurements","title":"Optimize Measurements","text":"","category":"section"},{"location":"Nonlocality/optimize_measurements/","page":"Optimize Measurements","title":"Optimize Measurements","text":"optimize_measurement","category":"page"},{"location":"Nonlocality/optimize_measurements/#BellScenario.Nonlocality.optimize_measurement","page":"Optimize Measurements","title":"BellScenario.Nonlocality.optimize_measurement","text":"LocalSignaling scenario:\n\noptimize_measurement(\n scenario :: LocalSignaling,\n game :: BellGame,\n ρ_states :: Vector{<:AbstractMatrix}\n)\n\nFinds the measurement that optimizes the score of the BellGame against the set of quantum states ρ_states. The optimization is performed with the following semi-definite program:\n\nbeginaligned\nmax_Pi_y sum_xy G_xy textTrPi_y rho_x \nst quad sum_y Pi_y = mathbbI quad Pi_y geq 0\nendaligned\n\n\n\n\n\nBipartiteNonSignaling scenario:\n\noptimize_measurement(\n game :: BellGame,\n scenario :: BipartiteNonSignaling,\n ρ_AB :: AbstractMatrix;\n A_POVMs :: Vector{<:AbstractVector{<:AbstractMatrix}},\n) :: Dict\n\nFind Bob's measurement which optimizes a BellGame's score for the shared quantum state ρ_AB and POVM measurement applied by Alice. The following semi-definite program optimizes the Bob's POVM:\n\nbeginaligned\n max_Pi_b^y sum_abxy G_abxytextTr(Pi_a^x otimes Pi_b^y)rho_AB \n st quad sum_b Pi_b^y = mathbbIquad Pi_b^y geq 0 quad forall y\nendaligned\n\noptimize_measurement(\n game :: BellGame,\n scenario :: BipartiteNonSignaling,\n ρ_AB :: AbstractMatrix;\n B_POVMs :: Vector{<:AbstractVector{<:AbstractMatrix}},\n) :: Dict\n\nFind Alice's measurement which optimizes a BellGame's score for the shared quantum state ρ_{AB} and POVM measurement applied by Bob. The following semi-definite program optimizes the Alice's POVM:\n\nbeginaligned\nmax_Pi_a^x sum_abxy G_abxytextTr(Pi_a^x otimes Pi_b^y)rho_AB \nst quad sum_a Pi_a^x = mathbbIquad Pi_a^x geq 0 quad forall x\nendaligned\n\n\n\n\n\n","category":"function"},{"location":"Nonlocality/overview/#Nonlocality.jl-Overview","page":"Overview","title":"Nonlocality.jl Overview","text":"","category":"section"},{"location":"Nonlocality/overview/","page":"Overview","title":"Overview","text":"Nonlocality","category":"page"},{"location":"Nonlocality/overview/#BellScenario.Nonlocality","page":"Overview","title":"BellScenario.Nonlocality","text":"Find the optimal parameters for quantum Bell violations.\n\nThe Nonlocality.jl module provides tools to optimize quantum non-locality in Bell scenarios.\n\nModule Exports:\n\noptimize_measurement: Finds the quantum measurement which violates a specified Bell inequality maximally.\n\n\n\n\n\n","category":"module"},{"location":"LocalPolytope/adjacency_decomposition/","page":"Adjacency Decomposition","title":"Adjacency Decomposition","text":"CurrentModule = LocalPolytope","category":"page"},{"location":"LocalPolytope/adjacency_decomposition/#Adjacency-Decomposition","page":"Adjacency Decomposition","title":"Adjacency Decomposition","text":"","category":"section"},{"location":"LocalPolytope/adjacency_decomposition/","page":"Adjacency Decomposition","title":"Adjacency Decomposition","text":"adjacency_decomposition\nadjacent_facets\nrotate_facet","category":"page"},{"location":"LocalPolytope/adjacency_decomposition/#BellScenario.LocalPolytope.adjacency_decomposition","page":"Adjacency Decomposition","title":"BellScenario.LocalPolytope.adjacency_decomposition","text":"adjacenecy_decomposition(\n vertices :: Vector{Vector{Int64}},\n BG_seed :: BellGame,\n scenario :: LocalSignaling;\n kwargs...\n) :: Dict\n\nGiven a polytpe represented by vertices, returns the complete set of canonical facets for prepare and measure scenario scenario. The adjacencydecomposition algorithm requires a seeded vertex which is supplied with the `BGseed` argument. Facets are returned in the lexicographic normal form.\n\nReturned Dictionary Format\n\nReturns a dictionary where the keys are canonical BellGames and the value is a dictionary with keys\n\n\"considered\" => true, if the facet was considered in the algorithm.\n\"skipped\" => true, if the facet was skipped (not considered).\n\"num_vertices\" => number of vertices.\n\"norm_facet\" => facet vector representative (normalized rep) of canonical game\n\nKeyword Arguments: kwargs...\n\nskip_games = [] :: Vector{BellGame} - List of games to skip.\nmax_vertices = 100 :: Int64 - The maximum number of vertices to allow in target facets.\ndir = \"./\" :: String- Directory in which to createporta_tmp/and.json` files.\nlog = false :: Bool - If true, the facet dictionary is written to .json each iteration.\nlog_filename = \"adjacency_decomposition_now.json\" :: String\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/adjacency_decomposition/#BellScenario.LocalPolytope.adjacent_facets","page":"Adjacency Decomposition","title":"BellScenario.LocalPolytope.adjacent_facets","text":"adjacent_facets(\n vertices :: Vector{Vector{Int64}},\n F :: Vector{int64};\n dir = \"./\" :: String,\n cleanup = true ::Bool\n) :: Vector{Vector{Int64}}\n\nFor the polytope represented by vertices, returns the set of facets adjacent to F.\n\nFacet vector F and the return facet vectors are assumed to ba in the normalized subspace.\n\nThe dir argument specifies where to where to write files and directories from XPORTA.jl. If cleanup is true, then a porta_tmp directory is created as a subdirectory of dir.\n\nIf cleanup is false, the created porta_tmp directory is not removed.\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/adjacency_decomposition/#BellScenario.LocalPolytope.rotate_facet","page":"Adjacency Decomposition","title":"BellScenario.LocalPolytope.rotate_facet","text":"rotate_facet(\n F :: Vector{Int64},\n G :: Vector{Int64},\n xbar :: Vector{Int64}\n) :: Vector{Int64}\n\nPerforms a rotation of facet F relative to non-included vertex xbar and returns the rotated facet vector. F is a polytope facet, G is a subfacet of F and xbar is a vertex not contained by F. By construction, the rotated facet contains xbar.\n\n\n\n\n\n","category":"function"},{"location":"user_guide/#User-Guide","page":"User Guide","title":"User Guide","text":"","category":"section"},{"location":"user_guide/#Quickstart","page":"User Guide","title":"Quickstart","text":"","category":"section"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"julia> using Pkg; Pkg.add(\"BellScenario\")","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"using BellScenario","category":"page"},{"location":"user_guide/#CHSH-Scenario","page":"User Guide","title":"CHSH Scenario","text":"","category":"section"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"The CHSH scenario is a BipartiteNonSignaling scenario where Alice and Bob each have a black-box with binary inputs and outputs.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"(Image: Classical CHSH Scenario)","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"This scenario is significant because it is the simplest Bell scenario in which quantum nonlocality can be observed. We will use BellScenario.jl to compute the CH Bell inequality and optimize quantum measurements to violate the CH inequality. First, create a CHSH Scenario to specify the black-box arrangement in the figure above.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"# (num_out_A, num_out_B, num_in_A, num_in_B)\nchsh_scenario = BipartiteNonSignaling(2,2,2,2)","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"Bell inequalities bound the set of local (classical) correlations. The local set is a convex polytope referred to as the local polytope and the facets of the local polytope are Bell inequalities. The standard method of computing Bell inequalities is to first compute the local polytope vertices, then apply a polytope transformation algorithm to compute the Bell inequalities.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"The BellScenario.jl package provides the LocalPolytope module to compute Bell inequalities. The first step is to commpute the vertex representation for the CHSH scenario.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"chsh_polytope = LocalPolytope.vrep(chsh_scenario)","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"Then, the Bell inequalities can computed using the LocalPolytope.facets function.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"chsh_facets = LocalPolytope.facets(chsh_polytope)","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"We'll take 15^th facet as it represents the CH inequality","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"- P_A(12) - P_B(11) + P(1111) - P(1112) + P(1121) + P(1122) leq 0","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"In fact, this inequality is equivalent to the more celebrated CHSH inequality. The difference is that the CH inequality is expressed in terms of probabilities whereas the CHSH inequality is expressed in terms of bipartite correlators.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"ch_inequality = chsh_facets[15]","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"Now that we have computed a Bell inequality, we can find a quantum violation using the Nonlocality module. In this example, we will fix Alice's measurement and the quantum state shared between Alice and Bob.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"# maximally entangled state\nρ_AB = [1 0 0 1;0 0 0 0;0 0 0 0;1 0 0 1]/2\n\n# Alice's measurement bases\nΠ_ax = [\n [[1 0;0 0], [0 0;0 1]], # Pauli Z basis\n [[1 1;1 1]/2, [1 -1;-1 1]/2] # Pauli X basis\n]","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"Then, we convert the ch_inequality into a general representation of a BellGame.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"ch_game = convert(BellGame, ch_inequality, chsh_scenario)","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"Finally, we optimize Bob's measurement with respect to the fixed state and measurements.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"opt_dict = Nonlocality.optimize_measurement(\n chsh_scenario, ch_game, ρ_AB, A_POVMs=Π_ax\n)","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"We see that the inequality is violated for the optimized measurement and states.","category":"page"},{"location":"user_guide/","page":"User Guide","title":"User Guide","text":"opt_dict[\"violation\"]","category":"page"},{"location":"BellScenario/games/","page":"Games","title":"Games","text":"CurrentModule = BellScenario","category":"page"},{"location":"BellScenario/games/#BellScenario.jl-Games","page":"Games","title":"BellScenario.jl - Games","text":"","category":"section"},{"location":"BellScenario/games/","page":"Games","title":"Games","text":"AbstractGame\nGame\nBellGame","category":"page"},{"location":"BellScenario/games/#BellScenario.AbstractGame","page":"Games","title":"BellScenario.AbstractGame","text":"An AbstractGame is the abstract type that is parent to type representing a cost function for strategies. A Game is a linear inequality dual to Strategy matrices, it is represented by a matrix G containing scalar coefficients and a bound beta. The bound of the linear inequality is typically a maximum score attainable by Bell scenario using a certain set of resources. A Game G is played by a Strategy S to achieve a score computed as,\n\nbeta geq langle G Srangle = sum_xy G_yxS(yx)\n\nThe game is \"won\" if the strategy scores greater than beta, that is, the above inequality is violated. Since beta is the maximum score for a set of resources, the game is won only if the tested strategy used a set of resources of greater operational value than than considered when computing the bound beta.\n\nConveniently, Strategy matrices have normalized columns and non-negative elements. This means that any game inequality can be converted into a form where game matrix G has positive elements and beta designates a positive upper bound.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/games/#BellScenario.Game","page":"Games","title":"BellScenario.Game","text":"Game(game::Matrix{T}, β::Real) <: AbstractGame{T}\n\nA Game is represented by a Matrix of coefficients and a scalar bound β.\n\nType parameter T can be either Int64 or Float64.\n\n\n\n\n\n","category":"type"},{"location":"BellScenario/games/#BellScenario.BellGame","page":"Games","title":"BellScenario.BellGame","text":"BellGame(game::Matrix{Int64}, β::Int64, scenario::Scenario)\n\nA BellGame represents a Bell inequality or tight facet of the local polytope. Since the vertices of the local polytope are deterministic strategies with 0,1 elements, the linear inequalities describing facets of the local polytope have rational coefficients. Therefore, if a inequality tightly bounds the local polytope, it can be represented by a game with integer coefficients. \n\n\n\n\n\n","category":"type"},{"location":"BellScenario/games/#Conversion-Methods","page":"Games","title":"Conversion Methods","text":"","category":"section"},{"location":"BellScenario/games/","page":"Games","title":"Games","text":"A BellGame is a matrix representation of a linear inequality that bounds a LocalPolytope for some Scenario. For convenience, conversion methods are provided to transform BellGames to alternative representations of polytope facets. There are two supported types which can be converted to/from BellGames:","category":"page"},{"location":"BellScenario/games/","page":"Games","title":"Games","text":"Facet - Vector{Int64}, a column-major vectorization of the game matrix with the bound placed in the last element of the vector.\nIEQ - A facet data structure defined in XPORTA.jl.","category":"page"},{"location":"BellScenario/games/","page":"Games","title":"Games","text":"convert(::Type{BellGame},::Vector{Int64},::BlackBox)\nconvert(::Type{BellGame},::Vector{Int64},::BipartiteNonSignaling)\nconvert(::Type{Vector{Int64}}, ::BellGame)\nconvert(::Type{Vector{Int64}},BG::BellGame,scenario::BipartiteNonSignaling)\nconvert(::Type{Vector{BellGame}},::IEQ,::BlackBox)\nconvert(::Type{IEQ}, bell_games::Vector{BellGame})","category":"page"},{"location":"BellScenario/games/#Base.convert-Tuple{Type{BellGame}, Vector{Int64}, BlackBox}","page":"Games","title":"Base.convert","text":"Facet (Vector{Int64}) -> BellGame\n\nconvert(\n ::Type{BellGame},\n facet::Vector{Int64},\n scenario::Union{BlackBox, LocalSignaling};\n rep = \"normalized\"::String\n)\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/games/#Base.convert-Tuple{Type{BellGame}, Vector{Int64}, BipartiteNonSignaling}","page":"Games","title":"Base.convert","text":"Facet (Vector{Int64}) -> BellGame\n\nconvert(\n ::Type{BellGame},\n facet::Vector{Int64},\n scenario::BipartiteNonSignaling;\n rep = \"non-signaling\"::String\n)\n\nTransforms LocalPolytope facets into BellGame types.\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/games/#Base.convert-Tuple{Type{Vector{Int64}}, BellGame}","page":"Games","title":"Base.convert","text":"BellGame -> Facet (Vector{Int64})\n\nconvert(::Type{Vector{Int64}}, BG::BellGame; rep = \"normalized\" :: String)\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/games/#Base.convert-Tuple{Type{Vector{Int64}}, BellGame, BipartiteNonSignaling}","page":"Games","title":"Base.convert","text":"BellGame -> Vector{Int64}\n\nconvert(::Type{Vector{Int64}},\n BG::BellGame,\n scenario::BipartiteNonSignaling;\n rep = \"non-signaling\" :: String\n)\n\nTransforms a BellGame for a BipartiteNonSignaling scenario into a facet vector.\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/games/#Base.convert-Tuple{Type{Vector{BellGame}}, XPORTA.IEQ, BlackBox}","page":"Games","title":"Base.convert","text":"XPORTA.IEQ to BellGame's\n\nconvert(\n ::Type{Vector{BellGame}},\n ieq::IEQ,\n scenario::Union{BlackBox, LocalSignaling};\n rep = \"normalized\" :: String\n)\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/games/#Base.convert-Tuple{Type{XPORTA.IEQ}, Vector{BellGame}}","page":"Games","title":"Base.convert","text":"BellGame's to XPORTA.IEQ\n\nconvert(::Type{IEQ}, bell_games::Vector{BellGame}; rep = \"normalized\" :: String)\n\n\n\n\n\n","category":"method"},{"location":"BellScenario/games/#File-I/O","page":"Games","title":"File I/O","text":"","category":"section"},{"location":"BellScenario/games/","page":"Games","title":"Games","text":"In practice, one may need to view are large set of BellGames in a human-readable form.","category":"page"},{"location":"BellScenario/games/","page":"Games","title":"Games","text":"pretty_print_txt","category":"page"},{"location":"BellScenario/games/#BellScenario.pretty_print_txt","page":"Games","title":"BellScenario.pretty_print_txt","text":"pretty_print_txt( bell_games :: Vector{BellGame}, filename :: String )\n\nPrints a set of BellGame's to filename.txt in a human-readable form.\n\n\n\n\n\n","category":"function"},{"location":"LocalPolytope/overview/#Local-Polytope-Overview","page":"Overview","title":"Local Polytope Overview","text":"","category":"section"},{"location":"LocalPolytope/overview/","page":"Overview","title":"Overview","text":"LocalPolytope","category":"page"},{"location":"LocalPolytope/overview/#BellScenario.LocalPolytope","page":"Overview","title":"BellScenario.LocalPolytope","text":"Compute the bounds of classical Bell scenarios.\n\nThe LocalPolytope.jl module provides tools for characterizing the convex polyhedral structure of the correlations attainable using classical resources in a Bell scenario.\n\nEach distinct strategy for a given Bell scenario corresponds to a point in vector space. The complete set of attainable strategies for a particular Bell scenario is denoted mathbfP. Given the constraints of normalization and non-negativity, the extreme points of mathbfP correspond to deterministic strategies (matrices with 0,1 elements). Furthermore, when shared randomness is used in a Bell scenario, any two strategies can be mixed together in a convex combination. Hence, mathbfP is a convex polyhedron referred to as the local polytope.\n\nA convex polytope has two equivalent descriptions:\n\nV-Description: The polytope is the convex hull of a set of extreme points known as vertices mathbfV, quadmathbfP = textconv(mathbfV).\nH-Description: The polytope is the intersection of a set of linear half-spaces known as facets mathbfF, quadmathbfP = capmathbfF.\n\nThese two descriptions are equivalent,\n\nmathbfP = textconv(mathbfV) = capmathbfF\n\nand there exists a transformation between the set of vertices and facets mathbfV leftrightarrow mathbfF.\n\nFor a given Bell scenario, the V-Description is typically easier to compute, however, the H-Description describes the bounds of the local polytope as as linear inequalities known as Bell inequalities. Bell inequalities are important because they provide a simple test to verify that a strategy is not contained by the local polyotpe. That is, if a Bell inequality is violated, the classical resources considered for the local polytope are not sufficient to reproduce the strategy. Hence, a Bell violation witnesses the use of resources with greater operational value. Bell violations are often used to characterize the advantages of quantum resources, however, it is important to note that a Bell violation can simply mean that more classical resources were used than anticipated.\n\nModule Exports:\n\nvertices - Compute the set of extreme-points for the local polytope.\nnum_vertices - The number of vertices for the local polytope.\nvrep - Construct a Polyhedron in the vertex representation.\nfacets - Compute the linear inequalities which bound the local polytope.\ngenerator_vertex - Provide a canonical form for a vertex.\ngenerator_facet - Provide a canonical form for a facet.\nadjacency_decomposition - Efficiently compute the generator facets for the local polytope using the adjacency decomposition technique.\n\n\n\n\n\n","category":"module"},{"location":"#BellScenario.jl","page":"Home","title":"BellScenario.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Compute Bell inequalities and their quantum violations.","category":"page"},{"location":"","page":"Home","title":"Home","text":"This software is a general tool for analyzing quantum non-locality in Bell scenarios.","category":"page"},{"location":"","page":"Home","title":"Home","text":"note: Alpha Version\nBreaking changes will be made in future commits.","category":"page"},{"location":"#Featured-Modules","page":"Home","title":"Featured Modules","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"BellScenario: Types and constructors for describing Bell scenarios.\nLocalPolytope: Computes the bounds classical Bell Scenarios.\nNonlocality: Optimizes quantum states and measurements for non-locality.","category":"page"},{"location":"#Overview","page":"Home","title":"Overview","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A Bell scenario describes a system of inter-connected black-box devices. A black-box is a simple device which takes an input and produces an output, however, no assumptions are made about the physical process inside the black-box. Black-boxes and Bell scenarios are completely characterized by their input-output statistics. The statistics of a Bell scenario depends on the underlying physics used by the black-box devices.","category":"page"},{"location":"","page":"Home","title":"Home","text":"In most familiar cases, a black-box uses classical physics. Some examples of classical black-boxes include cars, slot machines, and software. Classical physics is deterministic, although, an observer of a classical black-box may not have access to the parameters required to predict the output. These parameters are called local hidden variables where local means that events occurring in one location of space-time do not affect another location. Essentially, classical black-boxes are deterministic while their local hidden variables create the perception of probabilistic behavior. Hence, any probabilistic mixture of deterministic black-box behaviors can, in principle, be realized for a given Bell scenario. The bounds of a classical Bell scenario's statistics are regarded as Bell inequalities and the set of all possible behaviors forms a convex polyhedron known as the local polytope.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Bell scenarios are of interest to fundamental physics because not all physical systems adhere to the constraint of locality. For example, quantum Bell scenarios are able to realize statistics unattainable by their corresponding classical systems. This non-local behavior manifests as a strong correlation between distant points in space-time. The non-local correlations of quantum systems result in operational advantages in communications, information security, and data processing applications.","category":"page"},{"location":"","page":"Home","title":"Home","text":"While the success of many future technologies depends on realizing quantum non-locality, this task is very difficult, even in a theoretical sense. To witness non-locality, the local bounds must first be derived. Then, the parameters of the equivalent quantum system must be tuned to violate the derived Bell inequalities. Each of these problems are challenging analytically and numerically. The BellScenario.jl package provides the basic toolkit for finding non-locality in quantum systems.","category":"page"},{"location":"#Contents","page":"Home","title":"Contents","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Pages = [\"user_guide.md\", \"BellScenario/overview.md\", \"LocalPolytope/overview.md\", \"Nonlocality/overview.md\", \"development_manual.md\",]\nDepth = 1","category":"page"},{"location":"#Acknowledgements","page":"Home","title":"Acknowledgements","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Development of BellScenario.jl was made possible by the advisory of Dr. Eric Chitambar and general support from the Physics Department at the University of Illinois Urbana-Champaign. Funding was provided by NSF Award 1914440.","category":"page"},{"location":"#Citing","page":"Home","title":"Citing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To reference this work, see CITATION.bib.","category":"page"},{"location":"#Licensing","page":"Home","title":"Licensing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"BellScenario.jl is released under the MIT License.","category":"page"}] +} diff --git a/v0.1.3/siteinfo.js b/v0.1.3/siteinfo.js new file mode 100644 index 0000000..2500393 --- /dev/null +++ b/v0.1.3/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.1.3"; diff --git a/v0.1.3/user_guide/index.html b/v0.1.3/user_guide/index.html new file mode 100644 index 0000000..a5b874e --- /dev/null +++ b/v0.1.3/user_guide/index.html @@ -0,0 +1,71 @@ + +User Guide · BellScenario.jl

User Guide

Quickstart

julia> using Pkg; Pkg.add("BellScenario")
using BellScenario

CHSH Scenario

The CHSH scenario is a BipartiteNonSignaling scenario where Alice and Bob each have a black-box with binary inputs and outputs.

Classical CHSH Scenario

This scenario is significant because it is the simplest Bell scenario in which quantum nonlocality can be observed. We will use BellScenario.jl to compute the CH Bell inequality and optimize quantum measurements to violate the CH inequality. First, create a CHSH Scenario to specify the black-box arrangement in the figure above.

# (num_out_A, num_out_B, num_in_A, num_in_B)
+chsh_scenario = BipartiteNonSignaling(2,2,2,2)
BipartiteNonSignaling(2, 2, 2, 2)

Bell inequalities bound the set of local (classical) correlations. The local set is a convex polytope referred to as the local polytope and the facets of the local polytope are Bell inequalities. The standard method of computing Bell inequalities is to first compute the local polytope vertices, then apply a polytope transformation algorithm to compute the Bell inequalities.

The BellScenario.jl package provides the LocalPolytope module to compute Bell inequalities. The first step is to commpute the vertex representation for the CHSH scenario.

chsh_polytope = LocalPolytope.vrep(chsh_scenario)
Polyhedron XPORTA.Polyhedron:
+16-element iterator of Vector{Rational{Int64}}:
+ Rational{Int64}[1//1, 1//1, 1//1, 1//1, 1//1, 1//1, 1//1, 1//1]
+ Rational{Int64}[1//1, 1//1, 0//1, 1//1, 0//1, 1//1, 0//1, 1//1]
+ Rational{Int64}[1//1, 1//1, 1//1, 0//1, 1//1, 0//1, 1//1, 0//1]
+ Rational{Int64}[1//1, 1//1, 0//1, 0//1, 0//1, 0//1, 0//1, 0//1]
+ Rational{Int64}[0//1, 1//1, 1//1, 1//1, 0//1, 0//1, 1//1, 1//1]
+ Rational{Int64}[0//1, 1//1, 0//1, 1//1, 0//1, 0//1, 0//1, 1//1]
+ Rational{Int64}[0//1, 1//1, 1//1, 0//1, 0//1, 0//1, 1//1, 0//1]
+ Rational{Int64}[0//1, 1//1, 0//1, 0//1, 0//1, 0//1, 0//1, 0//1]
+ Rational{Int64}[1//1, 0//1, 1//1, 1//1, 1//1, 1//1, 0//1, 0//1]
+ Rational{Int64}[1//1, 0//1, 0//1, 1//1, 0//1, 1//1, 0//1, 0//1]
+ Rational{Int64}[1//1, 0//1, 1//1, 0//1, 1//1, 0//1, 0//1, 0//1]
+ Rational{Int64}[1//1, 0//1, 0//1, 0//1, 0//1, 0//1, 0//1, 0//1]
+ Rational{Int64}[0//1, 0//1, 1//1, 1//1, 0//1, 0//1, 0//1, 0//1]
+ Rational{Int64}[0//1, 0//1, 0//1, 1//1, 0//1, 0//1, 0//1, 0//1]
+ Rational{Int64}[0//1, 0//1, 1//1, 0//1, 0//1, 0//1, 0//1, 0//1]
+ Rational{Int64}[0//1, 0//1, 0//1, 0//1, 0//1, 0//1, 0//1, 0//1]

Then, the Bell inequalities can computed using the LocalPolytope.facets function.

chsh_facets = LocalPolytope.facets(chsh_polytope)
24-element Vector{Vector{Int64}}:
+ [0, 0, 0, 0, -1, 0, 0, 0, 0]
+ [0, 0, 0, 0, 0, -1, 0, 0, 0]
+ [0, 0, 0, 0, 0, 0, -1, 0, 0]
+ [0, 0, 0, 0, 0, 0, 0, -1, 0]
+ [-1, 0, 0, 0, 0, 1, 0, 0, 0]
+ [-1, 0, 0, 0, 1, 0, 0, 0, 0]
+ [0, -1, 0, 0, 0, 0, 0, 1, 0]
+ [0, -1, 0, 0, 0, 0, 1, 0, 0]
+ [0, 0, -1, 0, 0, 0, 1, 0, 0]
+ [0, 0, -1, 0, 1, 0, 0, 0, 0]
+ ⋮
+ [0, -1, 0, -1, -1, 1, 1, 1, 0]
+ [0, 1, 0, 1, 0, 0, 0, -1, 1]
+ [0, 1, 1, 0, 0, 0, -1, 0, 1]
+ [1, 0, 0, 1, 0, -1, 0, 0, 1]
+ [1, 0, 1, 0, -1, 0, 0, 0, 1]
+ [0, 1, 0, 1, 1, -1, -1, -1, 1]
+ [0, 1, 1, 0, -1, 1, -1, -1, 1]
+ [1, 0, 0, 1, -1, -1, 1, -1, 1]
+ [1, 0, 1, 0, -1, -1, -1, 1, 1]

We'll take $15^{th}$ facet as it represents the CH inequality

\[- P_A(1|2) - P_B(1|1) + P(11|11) - P(11|12) + P(11|21) + P(11|22) \leq 0.\]

In fact, this inequality is equivalent to the more celebrated CHSH inequality. The difference is that the CH inequality is expressed in terms of probabilities whereas the CHSH inequality is expressed in terms of bipartite correlators.

ch_inequality = chsh_facets[15]
9-element Vector{Int64}:
+  0
+ -1
+ -1
+  0
+  1
+ -1
+  1
+  1
+  0

Now that we have computed a Bell inequality, we can find a quantum violation using the Nonlocality module. In this example, we will fix Alice's measurement and the quantum state shared between Alice and Bob.

# maximally entangled state
+ρ_AB = [1 0 0 1;0 0 0 0;0 0 0 0;1 0 0 1]/2
+
+# Alice's measurement bases
+Π_ax = [
+    [[1 0;0 0], [0 0;0 1]],      # Pauli Z basis
+    [[1 1;1 1]/2, [1 -1;-1 1]/2] # Pauli X basis
+]
2-element Vector{Vector{Matrix{Float64}}}:
+ [[1.0 0.0; 0.0 0.0], [0.0 0.0; 0.0 1.0]]
+ [[0.5 0.5; 0.5 0.5], [0.5 -0.5; -0.5 0.5]]

Then, we convert the ch_inequality into a general representation of a BellGame.

ch_game = convert(BellGame, ch_inequality, chsh_scenario)
4×4 BellGame:
+ 1  0  1  1
+ 1  1  0  0
+ 0  1  1  0
+ 1  1  1  0

Finally, we optimize Bob's measurement with respect to the fixed state and measurements.

opt_dict = Nonlocality.optimize_measurement(
+    chsh_scenario, ch_game, ρ_AB, A_POVMs=Π_ax
+)
Dict{String, Any} with 7 entries:
+  "scenario"  => BipartiteNonSignaling(2, 2, 2, 2)
+  "A_POVMs"   => QBase.POVM{Float64}[[[1.0 0.0; 0.0 0.0], [0.0 0.0; 0.0 1.0]], …
+  "score"     => 3.20711
+  "game"      => [1 0 1 1; 1 1 0 0; 0 1 1 0; 1 1 1 0]
+  "violation" => 0.207109
+  "B_POVMs"   => QBase.POVM{ComplexF64}[[[0.853554+0.0im 0.353554+0.0im; 0.3535…
+  "state"     => Number[0.5 0.0 0.0 0.5; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0; 0.5 …

We see that the inequality is violated for the optimized measurement and states.

opt_dict["violation"]
0.20710899850467124
diff --git a/versions.js b/versions.js index d7dd1de..82a557e 100644 --- a/versions.js +++ b/versions.js @@ -3,5 +3,5 @@ var DOC_VERSIONS = [ "v0.1", "dev", ]; -var DOCUMENTER_NEWEST = "v0.1.2"; +var DOCUMENTER_NEWEST = "v0.1.3"; var DOCUMENTER_STABLE = "stable";