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<h2 class="hd hd-2 unit-title">Einstein, Podolsky, and Rosen's issues with quantum entanglement</h2>
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<h2 class="hd hd-2 unit-title">Bell's argument as a classical game between Alice and Bob</h2>
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<p>
See <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.23.880" target="_blank">Proposed Experiment to Test Local Hidden-Variable Theories</a>, by John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt (Phys. Rev. Lett. 23, 880, 1969). </p>
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<h2 class="hd hd-2 unit-title">Quantum protocol demonstrating EPR violation - analysis</h2>
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<h2 class="hd hd-2 unit-title">CQ: CHSH game strategy</h2>
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CHSH game strategy
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
In the CHSH game, as described by Prof. Peter Shor in lecture, Alice and Bob shared a specific two-qubit entangled state [mathjaxinline]|\psi \rangle[/mathjaxinline], such that given random [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline], without communicating, they are able to obtain [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] such that the probability of [mathjaxinline]x\oplus y = a\cdot b[/mathjaxinline] is [mathjaxinline]\cos ^2(\pi /8)\approx 0.854[/mathjaxinline]. This exceeds the maximum probability which can be obtained with any classical strategy, [mathjaxinline]0.75[/mathjaxinline]. </p>
<p>
Recall that the four Bell states are </p>
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[mathjaxinline]\displaystyle |\Phi ^+\rangle[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{|00\rangle + |11\rangle }{\sqrt{2}}[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.8)</td>
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<tr id="a0000000013">
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[mathjaxinline]\displaystyle |\Phi ^-\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{|00\rangle - |11\rangle }{\sqrt{2}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.9)</td>
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<tr id="a0000000014">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\Psi ^+\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{|01\rangle + |10\rangle }{\sqrt{2}}[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.10)</td>
</tr>
<tr id="a0000000015">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\Psi ^-\rangle[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle \frac{|01\rangle - |10\rangle }{\sqrt{2}} \, .[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.11)</td>
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<p>
Suppose Alice and Bob are given one of these Bell states as [mathjaxinline]|\psi \rangle[/mathjaxinline], and told which one they have. Say that some quantum protocol is &#8220;successful" when it allows Alice and Bob to win the CHSH game with probability [mathjaxinline]\cos ^2(\pi /8)[/mathjaxinline]. Does a successful protocol exist for every Bell state, none, or only one of them? </p>
<p>
Check each of the following statements which are true: </p>
<p>
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<text>A successful quantum protocol does NOT exist for any of the Bell states</text>
</label>
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<text>A successful quantum protocol exists only for [mathjaxinline]|\psi \rangle =|\Phi ^-\rangle[/mathjaxinline]</text>
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<text>A successful quantum protocol exists only for [mathjaxinline]|\psi \rangle =|\Psi ^+\rangle[/mathjaxinline]</text>
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<text>A successful quantum protocol exists only for [mathjaxinline]|\psi \rangle =|\Psi ^-\rangle[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">CQ: commuting matrices and simultaneous diagonalizability</h2>
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Commuting matrices and simultaneous diagonalizability
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<text>Commuting Hermitian matrices are simultaneously diagonalizable by some unitary matrix</text>
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<text>Commuting unitary matrices are simultaneously diagonalizable by some Hermitian matrix</text>
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<text>Any two matrices which commute are diagonalizable by some matrix</text>
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<text>Hermitian matrices which are simultaneously diagonalizable must commute with each other</text>
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<div id="U1L6j"></div>
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