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    "# Exercise 3"
   ]
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    "## 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."
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    "## 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",
    "$\n",
    "\\langle \\sigma^x \\rangle^2 + \\langle \\sigma^y \\rangle^2 + \\langle \\sigma^z \\rangle^2 = 1\n",
    "$\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",
    "$$\n",
    "p^z_0 - p^z_1 = \\langle \\psi | M | \\psi \\rangle \\,\\,\\, \\forall |\\psi\\rangle\n",
    "$$"
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