add ex 3

exercises/Exercise3.ipynb

@@ -0,0 +1,74 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Exercise 3\n",
+    "\n",
+    "This sheet is about [density matrices](../QI_course/2_The_Qubit.pdf), [the partial trace](../QI_course/3_Quantum_Information.pdf) and [the Schmidt decomposition](../QI_course/6_Quantum_Correlations_part_1.pdf). Take a look at the linked materials to find out more."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 1\n",
+    "\n",
+    "Given an abitrary single qubit state $a|0\\rangle +b|1\\rangle$:\n",
+    "\n",
+    "(a) Write down the correponding density matrix.\n",
+    "\n",
+    "(b) Write down the density matrix representing the effect of applying an $X$ with probability $q_x$, $Y$ with probability $q_y$ and $Z$ with probability $q_z$.\n",
+    "\n",
+    "(c) Write down the density matrix for representing the effect of replacing the state with $I/2$ with probability $p$.\n",
+    "\n",
+    "(d) For what values of $q_x$, $q_y$ and $q_z$ is the effect of (b) equivalent to that of (c)?\n",
+    "\n",
+    "## 2\n",
+    "\n",
+    "(a) Write down the density matrix of a single qubit which is in state $|0\\rangle$ with probability $2/3$ and $|+\\rangle$ with probability $1/3$.\n",
+    "\n",
+    "(b) Find a two qubit state such that one of the reduced density matrices is equal to the density matrix in (a).\n",
+    "\n",
+    "(c) Write down a density matrix for a state that is not entangled, but which has the same reduced density matrices as your answer for (b).\n",
+    "\n",
+    "## 3\n",
+    "\n",
+    "Consider the three qubit state $(|001\\rangle + |010\\rangle)+ |100\\rangle/\\sqrt{3}$.\n",
+    "\n",
+    "(a) What is the reduced density matrix for a single qubit in this state?\n",
+    "\n",
+    "(b) What is the reduced density matrix for a pair of qubits in this state?\n",
+    "\n",
+    "(c) Use the Schmidt decomposition to rewrite the state, with one qubit on one side of the bipartition and two qubits on the other."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
+  }
+ ],
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+   "language": "python",
+   "name": "python3"
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+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
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+   "file_extension": ".py",
+   "mimetype": "text/x-python",
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+}