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<p><big class="xlarge">More about the history of quantum teleporation - at IBM</big></p><p>
Read more about <a href="https://researcher.watson.ibm.com/researcher/view_group.php?id=2862" target="_blank">the history of quantum teleporation</a>, and how it was invented and first demonstrated by Charles Bennett, John Smolin, and colleagues at IBM in Yorktown. </p><html url_name="site_search_box1"/>
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<h2 class="hd hd-2 unit-title">CQ: Bell basis measurement circuit</h2>
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Bell basis measurement circuit
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
Suppose you wish to construct a quantum circuit such that measuring qubits [mathjaxinline]|ab\rangle[/mathjaxinline] of the four Bell states produce results [mathjaxinline]xy = 00, 01, 10, 11[/mathjaxinline], according to: </p>
<table id="a0000000002" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
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<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}} \longrightarrow xy = 00[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.1)</td>
</tr>
<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}} \longrightarrow xy = 10[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.2)</td>
</tr>
<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}} \longrightarrow xy = 01[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.3)</td>
</tr>
<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>
<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}} \longrightarrow xy = 11 \, .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.4)</td>
</tr>
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<p>
Note that these assigments of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] may be different from what you find elsewhere (e.g. in Nielsen &amp; Chuang); that is deliberate, so you think a bit harder about the circuit. </p>
<p>
Complete the following quantum circuit to implement the desired measurement, by placing the Hadamard gates ([mathjaxinline]H[/mathjaxinline]) and identity gates ([mathjaxinline]I[/mathjaxinline]) where appropriate, around the CNOT gate which has qubit [mathjaxinline]b[/mathjaxinline] as its control, and qubit [mathjaxinline]a[/mathjaxinline] as its target: </p>
<span>
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Teleportation circuit
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This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
Complete the following quantum circuit such that the output state [mathjaxinline]|\chi \rangle[/mathjaxinline] ends up being the input [mathjaxinline]|\psi \rangle[/mathjaxinline], and the two measured qubits give completely random classical outputs: </p>
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Superdense coding with three qubits
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
In lecture, Prof. Shor describes how superdense coding allows Alice to send two bits of information to Bob, by transferring just one qubit to Bob, assuming this qubit is part of a previously shared EPR pair of qubits. </p>
<p>
Suppose Alice and Bob share three qubits, such as the Greenberger-Horne-Zeilinger (&#8220;GHZ") state </p>
<table id="a0000000007" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \frac{|000\rangle + |111\rangle }{\sqrt{2}} \, ,[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.5)</td>
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with Alice holding the first qubit, and Bob the second and third qubits. </p>
<p>
How many bits of classical information can Alice send to Bob, by sending her single qubit to Bob, and why? </p>
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<text>Only one bit of classical information, because of Holevo's theorem</text>
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<text>Two bits of information, because it's just like superdense coding, despite the extra qubit</text>
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<text>Four bits of information, because Bob holds two qubits</text>
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<text>Eight bits of information, because three qubits can be in eight possible orthogonal states</text>
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