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Unitary and Hermitian matrices
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This is a concept question, provided to double-check your understanding of the previous video clip. </p>
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Which of the following statements about <a href="https://en.wikipedia.org/wiki/Unitary_matrix" target="_blank">unitary</a> and <a href="https://en.wikipedia.org/wiki/Hermitian_matrix" target="_blank">Hermitean matrices</a> are true? Check all that are valid: </p>
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<text>If [mathjaxinline]U[/mathjaxinline] is unitary, then it is also Hermitian</text>
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<text>If [mathjaxinline]U[/mathjaxinline] is unitary, then [mathjaxinline]U+U^\dagger[/mathjaxinline] is Hermitian</text>
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<text>If [mathjaxinline]H[/mathjaxinline] is Hermitian, then [mathjaxinline]H+H^\dagger[/mathjaxinline] is unitary</text>
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<h2 class="hd hd-2 unit-title">Two-qubit spin observables</h2>
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<h2 class="hd hd-2 unit-title">CQ: Two-qubit spin operators</h2>
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Two-qubit spin operators
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
Prof. Shor described in lecture how the total spin of two qubits can be measured using certain Hermitian matrix operators, composed using tensor products and addition. </p>
<p>
Drag the three labels below to match the correct matrix giving the corresponding Hermitian matrix representation: </p>
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$('div.course-index').find('nav').find('a').each(function(){
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<h2 class="hd hd-2 unit-title">CQ: Spin expectation values of the four Bell states</h2>
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Spin expectation values of the four Bell states
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
The four <a href="https://en.wikipedia.org/wiki/Bell_state" target="_blank">Bell states</a> are defined as </p>
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<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\Phi ^+\rangle[/mathjaxinline]
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<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]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.1)</td>
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<tr id="a0000000004">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[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]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.2)</td>
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<tr id="a0000000005">
<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 style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.3)</td>
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<tr id="a0000000006">
<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]
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[mathjaxinline]\displaystyle \frac{|01\rangle - |10\rangle }{\sqrt{2}} \, .[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.4)</td>
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<p>
In lecture, Prof. Shor also defined three total spin measurement operators: </p>
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<tr id="a0000000008">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle S_ x[/mathjaxinline]
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<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
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<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{X\otimes I + I \otimes X}{2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.5)</td>
</tr>
<tr id="a0000000009">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle S_ y[/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{Y\otimes I + I \otimes Y}{2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.6)</td>
</tr>
<tr id="a0000000010">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle S_ z[/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{Z\otimes I + I \otimes Z}{2} \, .[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.7)</td>
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<p>
Complete the following expressions giving the expectation values specified: </p>
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