Lecture_01.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ed475f94-e476-41b1-b5d5-020a2aecc109",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Author: Brent Artuch\n",
    "# 2024/09/16\n",
    "\n",
    "from sympy import *\n",
    "from sympy.physics.quantum import *\n",
    "from math import degrees\n",
    "from IPython.display import *\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "# plt.rcParams[\"text.usetex\"] = True"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "099a6a5a-19cd-4c00-bc0a-ad0f20b3a08e",
   "metadata": {},
   "source": [
    "## Complex Numbers \n",
    "Complex Numbers =  $\\mathbb{C}$ <br>\n",
    "Expressed as $a+bi$ st $a,b \\in \\mathbb{R}$ and $i=\\sqrt{-1}$ <br>\n",
    "Complex Conjugate: $(a+bi)^*$ = $a-bi$ <br>\n",
    "Sqaured Magnitude: $|a+bi|^2$ = $(a+bi)^*(a+bi)$ = $a^2 +b^2$ <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a4c84cab-26da-4e0c-aed3-49d9552057ab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 300x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( a + i b, \\  -1, \\  a - i b, \\  a^{2} + b^{2}, \\  3 - 2 i, \\  13\\right)$"
      ],
      "text/plain": [
       "⎛                       2    2             ⎞\n",
       "⎝a + ⅈ⋅b, -1, a - ⅈ⋅b, a  + b , 3 - 2⋅ⅈ, 13⎠"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a, b = symbols('a b')\n",
    "init_printing()\n",
    "\n",
    "# Expressed as:\n",
    "z = a + b*I\n",
    "# st a,b in R, and i is an indeterminate with:\n",
    "i_squared = I**2\n",
    "\n",
    "# Complex Conjugate: \n",
    "z_conjugate = a + (-1)*b*I\n",
    "\n",
    "# Squared Magnitude |a + bi|^2: \n",
    "z_squared_magnitude = expand(z * z_conjugate)\n",
    "\n",
    "# Example\n",
    "a_1, b_1 = 3, 2\n",
    "complex_number = a_1 + b_1*I\n",
    "conjugate_example = conjugate(complex_number)\n",
    "squared_magnitude_example = expand(complex_number * conjugate_example)\n",
    " \n",
    "# Plot the example \n",
    "plt.figure(figsize=(3,3))\n",
    "plt.title(\"Complex Plane 3 + 2i\")\n",
    "plt.annotate(r'$\\vec a$ + $\\vec bi$', xy=(3, 2))\n",
    "plt.annotate(r'$\\vec a$ = 3', xy=(1.5, 2.2))\n",
    "plt.annotate(r'$\\vec b$ = 2', xy=(3.2, 1))\n",
    "plt.arrow(0,0, a_1, b_1, head_width = 0.1, length_includes_head = True, width = 0.02) \n",
    "plt.arrow(3, 0, 0, b_1, head_width = 0.1, length_includes_head = True, width = 0.02, color = 'b', linestyle = '--')\n",
    "plt.arrow(0, 2, a_1, 0, head_width = 0.1, length_includes_head = True, width = 0.02, color = 'b', linestyle = '--')\n",
    "plt.xlim(0,4)\n",
    "plt.ylim(0,3)\n",
    "plt.ylabel('Imaginary') \n",
    "plt.xlabel('Real') \n",
    "plt.show() \n",
    "\n",
    "(z, i_squared, z_conjugate, z_squared_magnitude, conjugate_example, squared_magnitude_example)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7ddb0d2-4b0d-431a-9d03-191b2f5c5227",
   "metadata": {},
   "source": [
    "## Polar Form\n",
    "Let $z$ = $a+bi$ <br>\n",
    "$r$ = $||z||$ = $\\sqrt{a^2+b^2}$ <br>\n",
    "If $b$ = $0$, then $z$ = $||a||$ <br>\n",
    "The phase $\\varphi$ is called the argument. <br>\n",
    "$(r, \\varphi)$ represents a point in the complex plane. <br>\n",
    "$z$ = $r(\\cos\\varphi + i \\sin\\varphi)$ = $re^{i\\varphi}$ <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8d1b810b-0920-4d41-982a-183e2734ae16",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left( \\sqrt{a^{2} + b^{2}}, \\  \\left( r, \\  \\varphi\\right), \\  r \\left(i \\sin{\\left(\\varphi \\right)} + \\cos{\\left(\\varphi \\right)}\\right) = e^{i \\varphi} r, \\  \\sqrt{13} \\left(i \\sin{\\left(\\varphi \\right)} + \\cos{\\left(\\varphi \\right)}\\right) = \\sqrt{13} e^{i \\varphi}, \\  \\operatorname{atan}{\\left(\\frac{b}{a} \\right)}, \\  33.6900675259798\\right)$"
      ],
      "text/plain": [
       "⎛   _________                                                                                                              ↪\n",
       "⎜  ╱  2    2                                                   ⅈ⋅varphi                                             ⅈ⋅varp ↪\n",
       "⎜╲╱  a  + b  , (r, varphi), r⋅(ⅈ⋅sin(varphi) + cos(varphi)) = e        ⋅r, √13⋅(ⅈ⋅sin(varphi) + cos(varphi)) = √13⋅e       ↪\n",
       "⎝                                                                                                                          ↪\n",
       "\n",
       "↪                                ⎞\n",
       "↪ hi      ⎛b⎞                    ⎟\n",
       "↪   , atan⎜─⎟, 33.690067525979785⎟\n",
       "↪         ⎝a⎠                    ⎠"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi, r, e = symbols('varphi, r, e')\n",
    "# Calculate magnitude r using the Norm Sqaure or Pythagorean Theorem:\n",
    "r_calculated = sqrt(z_squared_magnitude)\n",
    "# Note: if b = 0, then r = ||a||\n",
    "\n",
    "# The phase phi is called the argument\n",
    "polar_coordinates = (r, phi)\n",
    "z_polar_form = Eq(r*(cos(phi) + I*sin(phi)), r*e**(I*phi))\n",
    "\n",
    "# Example \n",
    "r_example = sqrt(squared_magnitude_example)\n",
    "z_polar_form_example = Eq(r_example*(cos(phi) + I*sin(phi)), r_example*e**(I*phi))\n",
    "\n",
    "# Side Note: Calculating the angle phi\n",
    "phi_calculated = atan(b/a)\n",
    "\n",
    "# Example Continued\n",
    "rads_phi_example = np.atan(b_1/a_1)\n",
    "phi_example = degrees(rads_phi_example)\n",
    "sin_phi = np.sin(rads_phi_example)\n",
    "cos_phi = np.cos(rads_phi_example)\n",
    "\n",
    "# Plot \n",
    "radius = 1\n",
    "angles = np.linspace(0 * np.pi, 2 * np.pi, 100) \n",
    "xs = np.cos(angles)\n",
    "ys = np.sin(angles)\n",
    "\n",
    "plt.figure(figsize=(5,5))\n",
    "plt.title(\"Polar Coordinates 3 + 2i\")\n",
    "plt.annotate(r'$e^{i \\phi}$', xy=(0.85, 0.55), fontsize = 15)\n",
    "plt.annotate(r'$cos \\phi$', xy=(0.25, 0.65), fontsize = 15)\n",
    "plt.annotate(r'$sin \\phi$', xy=(0.95, 0.25), fontsize = 15)\n",
    "plt.annotate(r'$\\phi$', xy=(0.25, 0.05), fontsize = 15)\n",
    "plt.plot(xs, ys, color = 'black')\n",
    "plt.xlim(-radius-0.25, radius+0.25)\n",
    "plt.ylim(-radius-0.25, radius+0.25)\n",
    "plt.ylabel('Imaginary') \n",
    "plt.xlabel('Real') \n",
    "plt.gca().set_aspect('equal')\n",
    "plt.plot([radius+0.25, -radius-0.25], [0, 0], color = 'black')\n",
    "plt.plot([0, 0], [radius+0.25, -radius-0.25], color = 'black')\n",
    "plt.arrow(0, 0, cos_phi, sin_phi,  head_width = 0.1, length_includes_head = True, width = 0.02)\n",
    "plt.arrow(cos_phi, 0, 0, sin_phi, head_width = 0.1, length_includes_head = True, width = 0.02, color = 'b', linestyle = '--')\n",
    "plt.arrow(0, sin_phi, cos_phi, 0, head_width = 0.1, length_includes_head = True, width = 0.02, color = 'b', linestyle = '--')\n",
    "\n",
    "plt.show()\n",
    "\n",
    "(r_calculated, polar_coordinates, z_polar_form, z_polar_form_example, phi_calculated, phi_example)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26af0acb-0951-4a2f-ad49-45937898d693",
   "metadata": {},
   "source": [
    "## Unit of Information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7180a6a3-8253-4617-92e1-bfc9c190e7ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left( 0, \\  1\\right), \\  \\left( \\left[\\begin{matrix}1\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right]\\right), \\  \\left( \\left[\\begin{matrix}1\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right], \\  \\left[\\begin{matrix}c_{0}\\\\c_{1}\\end{matrix}\\right]\\right), \\  \\left[\\begin{matrix}c_{0}\\\\c_{1}\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\  \\left[\\begin{matrix}1\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛                                            ⎡√2⎤            ⎡√2⎤  ⎡ √2 ⎤  ⎡ √2 ⎤  ⎡  √2  ⎤⎞\n",
       "⎜                                            ⎢──⎥            ⎢──⎥  ⎢ ── ⎥  ⎢ ── ⎥  ⎢  ──  ⎥⎟\n",
       "⎜        ⎛⎡1⎤  ⎡0⎤⎞  ⎛⎡1⎤  ⎡0⎤  ⎡c₀⎤⎞  ⎡c₀⎤  ⎢2 ⎥  ⎡1⎤  ⎡0⎤  ⎢2 ⎥  ⎢ 2  ⎥  ⎢ 2  ⎥  ⎢  2   ⎥⎟\n",
       "⎜(0, 1), ⎜⎢ ⎥, ⎢ ⎥⎟, ⎜⎢ ⎥, ⎢ ⎥, ⎢  ⎥⎟, ⎢  ⎥, ⎢  ⎥, ⎢ ⎥, ⎢ ⎥, ⎢  ⎥, ⎢    ⎥, ⎢    ⎥, ⎢      ⎥⎟\n",
       "⎜        ⎝⎣0⎦  ⎣1⎦⎠  ⎝⎣0⎦  ⎣1⎦  ⎣c₁⎦⎠  ⎣c₁⎦  ⎢√2⎥  ⎣0⎦  ⎣1⎦  ⎢√2⎥  ⎢-√2 ⎥  ⎢√2⋅ⅈ⎥  ⎢-√2⋅ⅈ ⎥⎟\n",
       "⎜                                            ⎢──⎥            ⎢──⎥  ⎢────⎥  ⎢────⎥  ⎢──────⎥⎟\n",
       "⎝                                            ⎣2 ⎦            ⎣2 ⎦  ⎣ 2  ⎦  ⎣ 2  ⎦  ⎣  2   ⎦⎠"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Classical Bit \n",
    "c_bits = (0, 1)\n",
    "c_bit_vectors = (Matrix([[1], [0]]), Matrix([[0], [1]])) \n",
    "\n",
    "# Quantum Qubit\n",
    "c_0, c_1 = symbols(\"c_0, c_1\")\n",
    "qubits = (Matrix([[1], [0]]), Matrix([[0], [1]]), Matrix([[c_0], [c_1]])) \n",
    "# with |c_0|^2 + |c_1|^2 = 1, and |c_n|^2 is the probability of yielding state n\n",
    "\n",
    "# A qubit is a linear combination of basis states:\n",
    "q = c_0*qubits[0] + c_1*qubits[1]\n",
    "# st that c_0 and c_1 are complex in two dimensions \n",
    "\n",
    "# Example\n",
    "c_0e, c_1e = 1/sqrt(2), 1/sqrt(2)\n",
    "example_state = Matrix([[c_0e], [c_1e]])\n",
    "# when measured, yields 0 50% of the time and 1 50% of the time\n",
    "# Note: this is also the ket plus state and is a state of equal superposition\n",
    "\n",
    "# Side Note: Defining common states\n",
    "ket_0 = qubits[0]\n",
    "ket_1 = qubits[1]\n",
    "ket_plus = 1/sqrt(2)*(ket_0 + ket_1)\n",
    "ket_minus = 1/sqrt(2)*(ket_0 - ket_1)\n",
    "ket_i = 1/sqrt(2)*(ket_0 + I*ket_1)\n",
    "ket_minus_i = 1/sqrt(2)*(ket_0 - I*ket_1)\n",
    "\n",
    "(c_bits, c_bit_vectors, qubits, q, example_state, ket_0, ket_1, ket_plus, ket_minus, ket_i, ket_minus_i)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cfad633-3610-43e1-92c6-7efa24c75c02",
   "metadata": {},
   "source": [
    "## Storing More Information: Mulitiple Qubits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "8dd86666-baf7-414e-bfca-33ffd33c7c70",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}c_{00}\\\\c_{01}\\\\c_{10}\\\\c_{11}\\end{matrix}\\right], \\  \\left[\\begin{matrix}1\\\\0\\\\0\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\\\1\\\\0\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\\\0\\\\1\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\\\0\\\\0\\\\1\\end{matrix}\\right], \\  \\left(a_{0} {\\left|0\\right\\rangle } + a_{1} {\\left|1\\right\\rangle }\\right) \\left(c_{0} {\\left|0\\right\\rangle } + c_{1} {\\left|1\\right\\rangle }\\right), \\  a_{0} c_{0} {\\left|0\\right\\rangle }^{2} + a_{0} c_{1} {\\left|0\\right\\rangle } {\\left|1\\right\\rangle } + a_{1} c_{0} {\\left|1\\right\\rangle } {\\left|0\\right\\rangle } + a_{1} c_{1} {\\left|1\\right\\rangle }^{2}\\right)$"
      ],
      "text/plain": [
       "⎛⎡c₀₀⎤  ⎡1⎤  ⎡0⎤  ⎡0⎤  ⎡0⎤                                                                                              ⎞\n",
       "⎜⎢   ⎥  ⎢ ⎥  ⎢ ⎥  ⎢ ⎥  ⎢ ⎥                                                                                              ⎟\n",
       "⎜⎢c₀₁⎥  ⎢0⎥  ⎢1⎥  ⎢0⎥  ⎢0⎥                                                2                                            2⎟\n",
       "⎜⎢   ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, (a₀⋅❘0⟩ + a₁⋅❘1⟩)⋅(c₀⋅❘0⟩ + c₁⋅❘1⟩), a₀⋅c₀⋅❘0⟩  + a₀⋅c₁⋅❘0⟩⋅❘1⟩ + a₁⋅c₀⋅❘1⟩⋅❘0⟩ + a₁⋅c₁⋅❘1⟩ ⎟\n",
       "⎜⎢c₁₀⎥  ⎢0⎥  ⎢0⎥  ⎢1⎥  ⎢0⎥                                                                                              ⎟\n",
       "⎜⎢   ⎥  ⎢ ⎥  ⎢ ⎥  ⎢ ⎥  ⎢ ⎥                                                                                              ⎟\n",
       "⎝⎣c₁₁⎦  ⎣0⎦  ⎣0⎦  ⎣0⎦  ⎣1⎦                                                                                              ⎠"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Two Qubits\n",
    "c_00, c_01, c_10, c_11 = symbols(\"c_00, c_01, c_10, c_11\")\n",
    "q_12 = Matrix([[c_00], [c_01], [c_10], [c_11]])\n",
    "# st |c_OO|^2 + |c_01|^2 + |c_10|^2+ |c_11|^2 = 1, and |c_n|^2 is the probability of yielding state n\n",
    "\n",
    "# Defining two qubit states Ket 00, ket 01, ket 10, and ket 11 using the tensor product\n",
    "ket_00 = TensorProduct(ket_0, ket_0)\n",
    "ket_01 = TensorProduct(ket_0, ket_1)\n",
    "ket_10 = TensorProduct(ket_1, ket_0)\n",
    "ket_11 = TensorProduct(ket_1, ket_1)\n",
    "\n",
    "# A tensor product state:\n",
    "a_0, a_1 = symbols(\"a_0, a_1\")\n",
    "ket_zero, ket_one = Ket(\"0\"), Ket(\"1\")\n",
    "product_state = (a_0*ket_zero + a_1*ket_one) * (c_0*ket_zero + c_1*ket_one)\n",
    "expanded_product_state = expand(product_state)\n",
    "\n",
    "(q_12, ket_00, ket_01, ket_10, ket_11, product_state, expanded_product_state)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6d0b060-2514-4760-b05e-d735ac40f4b1",
   "metadata": {},
   "source": [
    "## Reading Out More Information: Mulitiple Qubits\n",
    "Consider the 2-Qubit system: $q_{1,2}$ = $c_{00}\\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}+c_{01}\\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{bmatrix}+c_{10}\\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{bmatrix}+c_{11}\\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}$ <br>\n",
    "st $|c_{00}|^2+|c_{01}|^2+|c_{10}|^2+|c_{11}|^2=1$ <br>\n",
    "Measure the first qubit to be zero with probability $|c_{00}|^2+|c_{01}|^2$ and post measurement state: <br>\n",
    "$q_1$ = $\\frac{1}{\\sqrt{|c_{00}|^2+|c_{01}|^2}} \\left(c_{00}\\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}+c_{01}\\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{bmatrix} \\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "e75a25b6-d4b3-449e-94cc-12e6c2614ab7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{c_{00}}{\\sqrt{\\left|{c_{00}}\\right|^{2} + \\left|{c_{01}}\\right|^{2}}}\\\\\\frac{c_{01}}{\\sqrt{\\left|{c_{00}}\\right|^{2} + \\left|{c_{01}}\\right|^{2}}}\\\\0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡        c₀₀         ⎤\n",
       "⎢────────────────────⎥\n",
       "⎢   _________________⎥\n",
       "⎢  ╱      2        2 ⎥\n",
       "⎢╲╱  │c₀₀│  + │c₀₁│  ⎥\n",
       "⎢                    ⎥\n",
       "⎢        c₀₁         ⎥\n",
       "⎢────────────────────⎥\n",
       "⎢   _________________⎥\n",
       "⎢  ╱      2        2 ⎥\n",
       "⎢╲╱  │c₀₀│  + │c₀₁│  ⎥\n",
       "⎢                    ⎥\n",
       "⎢         0          ⎥\n",
       "⎢                    ⎥\n",
       "⎣         0          ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Measure first qubit to be 0 from q_12\n",
    "q1 = (1/sqrt((abs(c_00))**2 + (abs(c_01))**2)) * (c_00*ket_00 + c_01*ket_01)\n",
    "# Note: The absolute value function is used simply to represent the norm for this case because their is no norm representation in sympy as far as I could find.\n",
    "\n",
    "(q1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64f099ec-6a6b-4b95-8902-900747ab2ad0",
   "metadata": {},
   "source": [
    "## Dirac Notation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7120e638-276e-468e-b1e7-134d3235335c",
   "metadata": {},
   "source": [
    "#### Ket notation (column vectors)\n",
    "$\\ket{0}$ = $\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}$ &emsp;\n",
    "$\\ket{1}$ = $\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}$ &emsp;\n",
    "$\\ket{q}$ = $\\begin{bmatrix} c_0 \\\\ c_1 \\end{bmatrix}$ = $c_0\\ket{0} + c_1\\ket{1}$\n",
    "\n",
    "#### Bra notation (conjugate row vectors)\n",
    "$\\Bra{0}$ = $\\begin{bmatrix} 1 & 0 \\end{bmatrix}$ &emsp;\n",
    "$\\Bra{1}$ = $\\begin{bmatrix} 0 & 1 \\end{bmatrix}$ &emsp;\n",
    "$\\Bra{q}$ = $\\begin{bmatrix} c_0^* & c_1^* \\end{bmatrix}$ = $c_0^*\\Bra{0} + c_1^*\\Bra{1}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b1c540b-0037-498e-ab7e-cf22ee3d25fc",
   "metadata": {},
   "source": [
    "## Braket / Inner Product \n",
    "$\\ket{\\psi}$ = $\\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix}$ &emsp;\n",
    "$\\ket{\\phi}$ = $\\begin{bmatrix} \\gamma \\\\ \\delta \\end{bmatrix}$ &emsp;\n",
    "$\\braket{\\psi|\\phi}$ = $\\begin{bmatrix} \\alpha^* & \\beta^* \\end{bmatrix}$  $\\begin{bmatrix} \\gamma \\\\ \\delta \\end{bmatrix}$ = $\\alpha^*\\gamma + \\beta^*\\delta$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f4e61379-a36e-4160-adc4-b496af207252",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left\\langle \\phi \\right. {\\left|\\psi\\right\\rangle }, \\  \\left[\\begin{matrix}c^{*}_{0} & c^{*}_{1} & c^{*}_{2}\\end{matrix}\\right], \\  \\left[\\begin{matrix}a_{0}\\\\a_{1}\\\\a_{2}\\end{matrix}\\right], \\  \\left[\\begin{matrix}a_{0} c^{*}_{0} + a_{1} c^{*}_{1} + a_{2} c^{*}_{2}\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛                                 ⎡a₀⎤                                          ⎞\n",
       "⎜                                 ⎢  ⎥                                          ⎟\n",
       "⎜⟨φ❘ψ⟩, [c_0__*  c_1__*  c_2__*], ⎢a₁⎥, [a₀⋅c_0__* + a₁⋅c_1__* + a₂⋅c_2__*], [0]⎟\n",
       "⎜                                 ⎢  ⎥                                          ⎟\n",
       "⎝                                 ⎣a₂⎦                                          ⎠"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Notation\n",
    "a_2 = symbols(\"a_2\")\n",
    "c_Oconj, c_1conj, c_2conj = symbols(\"c_0^* c_1^* c_2^*\")\n",
    "bra_phi = Bra(\"phi\")\n",
    "ket_psi = Ket(\"psi\")\n",
    "braket_phipsi = bra_phi * ket_psi\n",
    "\n",
    "# Inner Product\n",
    "bra_vector = transpose(Matrix([[c_Oconj],[c_1conj],[c_2conj]]))\n",
    "ket_vector = Matrix([[a_0],[a_1],[a_2]])\n",
    "\n",
    "inner_product = bra_vector * ket_vector\n",
    "# Results in a single scalar value\n",
    "\n",
    "# Example \n",
    "braket_bra0_ket1 = Dagger(ket_0) * ket_1 \n",
    "# Note: Dagger is the conjugate transpose of ket_0 giving us bra_0\n",
    "\n",
    "(braket_phipsi, bra_vector, ket_vector, inner_product, braket_bra0_ket1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e1383bc-e91d-4c77-aeaa-8fe3eac06fb3",
   "metadata": {},
   "source": [
    "## Orthonormality\n",
    "Let $\\ket{\\psi}$ = $\\alpha\\ket{0}+\\beta\\ket{1}$ <br>\n",
    "Take the inner product of $\\psi$ with itself to show that it is normalized: <br>\n",
    "$$\n",
    "\\braket{\\psi|\\psi} = \\begin{bmatrix} \\alpha^* & \\beta^* \\end{bmatrix}  \\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix} = |\\alpha^2|+|\\beta^2|=1\n",
    "$$ \n",
    "If the result of the inner product is zero for two states, then those two states are orthogonal as shown."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e2970b5e-3aee-422a-9ffe-327b8e3b5971",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}1\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\end{matrix}\\right], \\  \\left[\\begin{matrix}0\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "([1], [0], [0], [0])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Inner Product of a state with itself\n",
    "self_inner_product = Dagger(ket_0) * ket_0\n",
    "# The result is the total probability of 1, and that the state is normalized.\n",
    "\n",
    "# Inner product of basis states\n",
    "# Ket 0 and Ket 1\n",
    "inner_product_z_basis = Dagger(ket_0) * ket_1 \n",
    "# Ket + and Ket -\n",
    "inner_product_x_basis = Dagger(ket_plus) * ket_minus \n",
    "# Ket i and Ket -i\n",
    "inner_product_y_basis = Dagger(ket_i) * ket_minus_i\n",
    "# The result is 0, and therefore the basis states are orthogonal to their counterparts.\n",
    "# Since the basis states are normalized and orthogonal, they can be classified as orthonormal. \n",
    "\n",
    "(self_inner_product, inner_product_z_basis, inner_product_x_basis, inner_product_y_basis)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f501f99a-b065-4c75-82d8-17a82f5ba1d2",
   "metadata": {},
   "source": [
    "## Entanglement \n",
    "Two states are considered entangled if they cannot be written in the form: <br>\n",
    "$$\n",
    "\\ket\\psi_1\\ket{\\psi_0} = (\\alpha_1\\ket{0}+\\beta_1\\ket{1})\\otimes(\\alpha_0\\ket{0}+\\beta_0\\ket{1})\n",
    "= \\alpha_1\\alpha_0\\ket{00} + \\alpha_1\\beta_0\\ket{01} + \\beta_1\\alpha_0\\ket{10} + \\beta_1\\beta_0\\ket{11}\n",
    "$$\n",
    "So, if we have the bell state $\\ket{\\Phi^+}$ = $\\frac{1}{\\sqrt{2}}(\\ket{00}+\\ket{11})$, then the crresponding amplitudes are: \n",
    "$\\alpha_1\\alpha_0$ = $\\frac{1}{\\sqrt{2}}$, $\\alpha_1\\beta_0$ = 0, $\\beta_1\\alpha_0$ = 0,  $\\beta_1\\beta_0$ = $\\frac{1}{\\sqrt{2}}$ <br>\n",
    "We cannot solve this system of equations, thus this is an entagled state. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "978eacad-4ccc-4c6e-ab74-94b94e87e058",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}\\frac{1}{2}\\\\\\frac{1}{2}\\\\\\frac{1}{2}\\\\\\frac{1}{2}\\end{matrix}\\right], \\  \\frac{1}{4}, \\  \\frac{1}{4}, \\  \\frac{1}{4}, \\  \\frac{1}{4}, \\  \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\0\\\\0\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\  \\frac{1}{2}, \\  0, \\  0, \\  \\frac{1}{2}\\right)$"
      ],
      "text/plain": [
       "⎛                           ⎡√2⎤                ⎞\n",
       "⎜                           ⎢──⎥                ⎟\n",
       "⎜⎡1/2⎤                      ⎢2 ⎥                ⎟\n",
       "⎜⎢   ⎥                      ⎢  ⎥                ⎟\n",
       "⎜⎢1/2⎥                      ⎢0 ⎥                ⎟\n",
       "⎜⎢   ⎥, 1/4, 1/4, 1/4, 1/4, ⎢  ⎥, 1/2, 0, 0, 1/2⎟\n",
       "⎜⎢1/2⎥                      ⎢0 ⎥                ⎟\n",
       "⎜⎢   ⎥                      ⎢  ⎥                ⎟\n",
       "⎜⎣1/2⎦                      ⎢√2⎥                ⎟\n",
       "⎜                           ⎢──⎥                ⎟\n",
       "⎝                           ⎣2 ⎦                ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Consider the unentangeled states:\n",
    "psi_a = (1/sqrt(2))*(ket_0 + ket_1)\n",
    "psi_b = (1/sqrt(2))*(ket_0 + ket_1)\n",
    "\n",
    "# Take their tensor product \n",
    "psi_ab = TensorProduct(psi_a, psi_b)\n",
    "# Factor the coefficient for each state in the vector and take norm square\n",
    "probability_00 = psi_ab[0] * conjugate(psi_ab[0])\n",
    "probability_01 = psi_ab[1] * conjugate(psi_ab[1])\n",
    "probability_10 = psi_ab[2] * conjugate(psi_ab[2])\n",
    "probability_11 = psi_ab[3] * conjugate(psi_ab[3])\n",
    "# These states can be factored into a product state since they all have some probability of occuring\n",
    "\n",
    "# Let's say we have an entangled state\n",
    "entangled_psi_ab = (1/sqrt(2))*ket_00 + (1/sqrt(2))*ket_11\n",
    "# Factor the coefficient for each state in the vector and take norm square\n",
    "entangled_probability_00 = entangled_psi_ab[0] * conjugate(entangled_psi_ab[0])\n",
    "entangled_probability_01 = entangled_psi_ab[1] * conjugate(entangled_psi_ab[1])\n",
    "entangled_probability_10 = entangled_psi_ab[2] * conjugate(entangled_psi_ab[2])\n",
    "entangled_probability_11 = entangled_psi_ab[3] * conjugate(entangled_psi_ab[3])\n",
    "# Now we only have the possibility of the 00 and 11 states and the entangled state cannot be factored into a product state\n",
    "\n",
    "(psi_ab, probability_00, probability_01, probability_10, probability_11, entangled_psi_ab, entangled_probability_00,\n",
    " entangled_probability_01, entangled_probability_10, entangled_probability_11)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f01dec38-e3a4-47ee-b914-65ad91f858f9",
   "metadata": {},
   "source": [
    "## Orthogonal Measurment\n",
    "$\\ket{q}$ = $\\begin{bmatrix} c_0 \\\\ c_1 \\end{bmatrix}$ &emsp; Measure $\\ket{q}$ in the $\\ket{+}$ basis. <br>\n",
    "Recall $\\ket{+}$ = $\\begin{bmatrix} \\frac{1}{\\sqrt{2}} \\\\ \\frac{1}{\\sqrt{2}} \\end{bmatrix}$ = $\\frac{1}{\\sqrt{2}}(\\ket{0}+\\ket{1})$ <br>\n",
    "Thus, $\\ket{-}$ = $\\begin{bmatrix} \\frac{1}{\\sqrt{2}} \\\\ -\\frac{1}{\\sqrt{2}} \\end{bmatrix}$ = $\\frac{1}{\\sqrt{2}}(\\ket{0}-\\ket{1})$ <br>\n",
    "<br>\n",
    "We can now algebraically solve for $\\ket{0}$ in terms of the plus state: <br>\n",
    "$\\ket{+}$ = $\\frac{1}{\\sqrt{2}}(\\ket{0}+\\ket{1})$ <br>\n",
    "$\\sqrt{2}\\ket{+}$ = $\\ket{0}+\\ket{1}$ <br>\n",
    "$\\ket{0}$ = $\\sqrt{2}\\ket{+}-\\ket{1}$ = $\\sqrt{2}\\ket{+}-\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}$ = $\\sqrt{2}\\ket{+}+\\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix}$ * Add $\\ket{0}$ to both sides <br>\n",
    "$2\\ket{0}$ = $\\sqrt{2}\\ket{+}+\\begin{bmatrix} 1 \\\\ -1 \\end{bmatrix}$ * Multiply through by $\\frac{1}{\\sqrt{2}}$<br>\n",
    "$\\frac{2}{\\sqrt{2}}\\ket{0}$ = $\\ket{+}+\\ket{-}$ <br>\n",
    "$\\ket{0}$ = $\\frac{1}{\\sqrt{2}}(\\ket{+}+\\ket{-})$ <br>\n",
    "<br>\n",
    "Through a similar process, <br>\n",
    "$\\ket{1}$ = $\\frac{1}{\\sqrt{2}}(\\ket{+}-\\ket{-})$<br>\n",
    "<br>\n",
    "And we can describe a state $q$ in terms of $\\ket{+}$ and $\\ket{-}$ as: <br>\n",
    "$\\ket{q}$ = $\\begin{bmatrix} c_0 \\\\ c_1 \\end{bmatrix}$ = $c_0\\ket{0}+c_1\\ket{1}$ = $c_0\\frac{1}{\\sqrt{2}}(\\ket{+}+\\ket{-})+c_1\\frac{1}{\\sqrt{2}}(\\ket{+}-\\ket{-})$ = $\\frac{1}{\\sqrt{2}} ((c_0+c_1)\\ket{+}+(c_0+c_1)\\ket{-})$ <br>\n",
    "Probability of measuring $\\ket{+}$: $|\\frac{1}{\\sqrt{2}}(c_0+c_1)|^2$ = $\\frac{|c_0+c_1|^2}{2}$ <br>\n",
    "Probability of measuring $\\ket{-}$: $|\\frac{1}{\\sqrt{2}}(c_0-c_1)|^2$ = $\\frac{|c_0-c_1|^2}{2}$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "7b1d7a2a-9fbb-4902-a557-975125f0144e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Is Orthoganal: True\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\  \\frac{1}{2}, \\  \\frac{1}{2}\\right)$"
      ],
      "text/plain": [
       "⎛⎡√2⎤          ⎞\n",
       "⎜⎢──⎥          ⎟\n",
       "⎜⎢2 ⎥          ⎟\n",
       "⎜⎢  ⎥, 1/2, 1/2⎟\n",
       "⎜⎢√2⎥          ⎟\n",
       "⎜⎢──⎥          ⎟\n",
       "⎝⎣2 ⎦          ⎠"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example \n",
    "orthogonal = Dagger(ket_0) * ket_1 == Matrix([0])\n",
    "c_0 = c_1 = 1/sqrt(2)\n",
    "ket_q = c_0*ket_0 + c_1*ket_1\n",
    "prob_ket_0 = prob_ket_1 = c_0 * conjugate(c_0)\n",
    "\n",
    "print(\"Is Orthoganal:\", orthogonal)\n",
    "(ket_q, prob_ket_0, prob_ket_0)"
   ]
  }
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