From 6340f180e78512651583ae1fd29854d1342f84b1 Mon Sep 17 00:00:00 2001
From: James Wootton <decodoku@gmail.com>
Date: Tue, 22 Mar 2022 10:26:51 +0100
Subject: [PATCH] ex3

---
 exercises/Exercise3.ipynb | 87 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 87 insertions(+)
 create mode 100644 exercises/Exercise3.ipynb

diff --git a/exercises/Exercise3.ipynb b/exercises/Exercise3.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9ae539e2",
+   "metadata": {},
+   "source": [
+    "# Exercise 3"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fe84364c",
+   "metadata": {},
+   "source": [
+    "## 1. Mutually unbiased bases\n",
+    "Show that the $X$, $Y$ and $Z$ bases are all unbiased with respect to each other: each\n",
+    "state for one basis results in a completely random result for the other two bases."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7e5f84c8",
+   "metadata": {},
+   "source": [
+    "## 2. Shifting certainty\n",
+    "\n",
+    "The state of a single qubit is characterized by three numbers, $\\langle \\sigma^x \\rangle$, $\\langle \\sigma^y \\rangle$ and $\\langle \\sigma^z \\rangle$,\n",
+    "defined as\n",
+    "\n",
+    "$$\n",
+    "\\langle \\sigma^\\alpha \\rangle = p^\\alpha_0 - p^\\alpha_1,\n",
+    "$$\n",
+    "\n",
+    "where $p^z_0$ is the probability of the outcome \\texttt{0} for a $Z$ measurement, and so on.\n",
+    "\n",
+    "For any single qubit superposition $|\\psi\\rangle = c_0 |0\\rangle + c_1 |1\\rangle$,\n",
+    "\n",
+    "$\n",
+    "\\langle \\sigma^x \\rangle^2 + \\langle \\sigma^y \\rangle^2 + c^2 = 1\n",
+    "$\n",
+    "\n",
+    "Verify this for the following states.\n",
+    "\n",
+    "* (a) $|\\psi\\rangle = \\cos \\theta \\, |0\\rangle + \\sin \\theta \\, |1\\rangle$\n",
+    "* (b) $|\\psi\\rangle = \\frac{1}{\\sqrt{2}} \\, \\left( |0\\rangle \\,+\\, e^{i \\phi} |1\\rangle \\right)$\n",
+    "\n",
+    "## 3. A useful matrix\n",
+    "\n",
+    "Find the $2\\times2$ matrix $M$ such that\n",
+    "\n",
+    "$$\n",
+    "p^z_0 - p^z_1 = \\langle \\psi | M | \\psi \\rangle \\,\\,\\, \\forall |\\psi\\rangle\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ac918ed2",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.1"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}