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+    "# Exercise 4\n",
+    "\n",
+    "\n",
+    "## 1. Alternative Pauli Basis States\n",
+    "\n",
+    "There are an infinite number of possible single qubit states.  From a theoretical stand-point,  the  one  we  choose  to  label $| 0 \\rangle$ is  arbitrary.   So  let’s  consider  the  following alternative.\n",
+    "\n",
+    "$$\n",
+    "| \\bar 0 \\rangle = \\cos(\\theta) \\, | 0 \\rangle + \\sin(\\theta) \\, | 1 \\rangle.\n",
+    "$$\n",
+    "\n",
+    "For this $| \\bar 0 \\rangle$:\n",
+    "\n",
+    "(a) Find a corresponding orthogonal state $| \\bar 1 \\rangle$;\n",
+    "\n",
+    "(b)  For this basis $| \\bar 0 \\rangle$, $| \\bar 1 \\rangle$, find mutually unbiased basis states $| \\bar + \\rangle$ and $| \\bar - \\rangle$;\n",
+    "\n",
+    "## 2. Properties of the Pauli Matrices\n",
+    "\n",
+    "Note: Sometimes the Pauli matrices are written as $X$, $Y$ and $Z$, and sometimes as $\\sigma_x$, $\\sigma_y$ and $\\sigma_z$. For  the  most  part,  the  convention  is  an  arbitrary  choice. Once  you’ve used them enough,  you’ll hardly notice the difference (to the great annoyance of your students!).\n",
+    "\n",
+    "The Pauli matrices are defined\n",
+    "\n",
+    "$$\n",
+    "X = \n",
+    "\\begin{pmatrix} \n",
+    "0 & 1 \\\\\n",
+    "1 & 0 \\\\\n",
+    "\\end{pmatrix}, \\,\\,\n",
+    "Y = \n",
+    "\\begin{pmatrix} \n",
+    "0 & -i \\\\\n",
+    "i & 0 \\\\\n",
+    "\\end{pmatrix}, \\,\\,\n",
+    "Z = \n",
+    "\\begin{pmatrix} \n",
+    "1 & 0 \\\\\n",
+    "0 & -1 \\\\\n",
+    "\\end{pmatrix}, \\,\\,\n",
+    "$$\n",
+    "\n",
+    "(a) Show that each squares to the identity matrix.\n",
+    "\n",
+    "$$\n",
+    "I = \n",
+    "\\begin{pmatrix} \n",
+    "1 & 0 \\\\\n",
+    "0 & 1 \\\\\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "(b) Show that $P_1 P_2 = - P_2 P_1$ for any pair of Paulis $P_1$ and $P_2$.\n",
+    "\n",
+    "(c) Show that $P_1 P_2 \\sim P_3$ for any pair of Paulis $P_1$ and $P_2$, where $P_3$ is the remaining Pauli.\n",
+    "\n",
+    "(d) Find the eigenvectors and eigenvalues of each Pauli.\n",
+    "\n",
+    "\n",
+    "## 3. The Hadamard\n",
+    "\n",
+    "The Hadamard matrix can be expressed\n",
+    "\n",
+    "$$\n",
+    "H = \\frac{1}{\\sqrt{2}}\n",
+    "\\begin{pmatrix} \n",
+    "1 & 1 \\\\\n",
+    "1 & -1 \\\\\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "(a) Find the eigenvectors and eigenvalues of this matrix.\n",
+    "\n",
+    "(b) Show that $H$ also squares to identity.\n",
+    "\n",
+    "(c) Show that $H P_1 H^\\dagger \\sim P_2$ for Paulis $P_1$ and $P_2$. \n",
+    "\n",
+    "\n",
+    "## 4. Two-qubit Paulis\n",
+    "\n",
+    "For two qubits we can define a set of matrices $X_0$, $Y_0$ and $Z_0$ that behave as Paulis (i.e. they have the same properties as in 3a, 3b and 3c above). We can also define another separate set of matrices $X_1$, $Y_1$ and $Z_1$ that also behave as Paulis. Furthermore, any matrix from one of these sets will commute with any matrix from the other\n",
+    "\n",
+    "$$\n",
+    "P_0 P_1 = P_1 P_0, \\,\\, \\forall \\,\\, P_j \\, \\in \\, \\{X_j, Y_j, Z_j\\}.\n",
+    "$$\n",
+    "\n",
+    "Usually we define these sets in a very simple way, using the Paulis of one qubit for one set, and the Paulis of the other qubit for the other set,\n",
+    "\n",
+    "$$\n",
+    "X_0 = X \\otimes I, \\, X_1 = I \\otimes X, \\, \\rm{etc}.\n",
+    "$$\n",
+    "\n",
+    "But since this is an exercise, let's do something a bit more interesting! Consider the following choice of the first set,\n",
+    "\n",
+    "$$\n",
+    "X_0 = X \\otimes X, \\, Y_0 = Y \\otimes X,\\, Z_0 = Z \\otimes I\n",
+    "$$\n",
+    "\n",
+    "Find a corresponding $X_1$, $Y_1$ and $Z_1$."
+   ]
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