<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Quantum_teleportation" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="vertical"> <h2 class="hd hd-2 unit-title">Quantum teleportation</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Quantum_teleportation"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Quantum_teleportation" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="problem"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_problem_Quantum_teleportation" class="problems-wrapper" role="group" aria-labelledby="problem_Quantum_teleportation-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Quantum_teleportation" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Quantum_teleportation/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="6" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;problem_Quantum_teleportation-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Quantum_teleportation-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Quantum teleportation &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Quantum_teleportation-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; This is the problem on a slightly modified version of the quantum teleportation protocol where we use a different Bell state than what was used in the lecture. &lt;/p&gt; &lt;div id=&#34;fig1&#34; class=&#34;figure&#34;&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/1d6cccef4a026cffdda4e863ba653f74/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_ps3a_bell.png&#34; width=&#34;400&#34; style=&#34;scale:0.8&#34;/&gt; &lt;/center&gt; &lt;div class=&#34;caption&#34;&gt;&lt;b&gt;Figure 1&lt;/b&gt;: &lt;span&gt;Bell measurement&lt;/span&gt;&lt;/div&gt; &lt;/div&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; Recall the four Bell states &lt;/p&gt; &lt;table id=&#34;a0000000010&#34; class=&#34;eqnarray&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000011&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle |{\Phi ^+}\rangle[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle \frac{1}{\sqrt{2}}(|{00}\rangle +|{11}\rangle ),\ \ \ |{\Phi ^-}\rangle =\frac{1}{\sqrt{2}}(|{00}\rangle -|{11}\rangle ),[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.6)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;a0000000012&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle |{\Psi ^+}\rangle[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle \frac{1}{\sqrt{2}}(|{01}\rangle +|{10}\rangle ),\ \ \ |{\Psi ^-}\rangle =\frac{1}{\sqrt{2}}(|{01}\rangle -|{10}\rangle )[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none;text-align:right&#34;&gt;(1.7)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; form an orthonormal basis for two-qubit states. &lt;/p&gt; &lt;p&gt; The circuit in the above figure is called Bell measurement because it allows for the measurement with [mathjaxinline]\{ |{\Phi ^+}\rangle , |{\Phi ^-}\rangle , |{\Psi ^+}\rangle , |{\Psi ^-}\rangle \}[/mathjaxinline] basis. What is the probability of getting the measurement result 01 for the state [mathjaxinline]|{\psi }\rangle =a|{\Phi ^+}\rangle +b|{\Phi ^-}\rangle +c|{\Psi ^+}\rangle +d|{\Psi ^-}\rangle[/mathjaxinline]? Note that measurement result 01 means that you obtain result 0 for the upper measurement and result 1 for the lower measurement in the figure. Input &lt;tt class=&#34;ttfamily&#34;&gt;abs(a)&lt;/tt&gt; for [mathjaxinline]|a|[/mathjaxinline] etc. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Quantum_teleportation_2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_problem_Quantum_teleportation_2_1&#34; id=&#34;input_problem_Quantum_teleportation_2_1&#34; aria-describedby=&#34;status_problem_Quantum_teleportation_2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_problem_Quantum_teleportation_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Quantum_teleportation_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_problem_Quantum_teleportation_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_problem_Quantum_teleportation_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_problem_Quantum_teleportation_2_1_dynamath&#34; name=&#34;input_problem_Quantum_teleportation_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;div id=&#34;fig2&#34; class=&#34;figure&#34;&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/81195af357659192232a46cf743fc9f4/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_ps3a_teleportation.png&#34; width=&#34;400&#34; style=&#34;scale:0.5&#34;/&gt; &lt;/center&gt; &lt;div class=&#34;caption&#34;&gt;&lt;b&gt;Figure 2&lt;/b&gt;: &lt;span&gt;Teleportation&lt;/span&gt;&lt;/div&gt; &lt;/div&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; The above figure shows the quantum teleportation protocol where [mathjaxinline]A[/mathjaxinline] has a quantum state [mathjaxinline]|{\psi }\rangle =a|{0}\rangle +b|{1}\rangle[/mathjaxinline] and [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] share the entangled state [mathjaxinline]|{\Psi ^-}\rangle[/mathjaxinline]. We take [mathjaxinline]a,b[/mathjaxinline] as real numbers. [mathjaxinline]A[/mathjaxinline]&#39;s goal is to send [mathjaxinline]|{\psi }\rangle[/mathjaxinline] to [mathjaxinline]B[/mathjaxinline] using only local operations and classical communications. Find the state of [mathjaxinline]B[/mathjaxinline] right after the Bell measurement when the measurement result for the Bell measurement is 00. Take the global phase so that the coefficients are real. Remember to include [mathjaxinline]*[/mathjaxinline] for the multiplication; input &lt;tt class=&#34;ttfamily&#34;&gt;a*|0&amp;gt;&lt;/tt&gt; for [mathjaxinline]a|{0}\rangle[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Quantum_teleportation_3_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_problem_Quantum_teleportation_3_1&#34; id=&#34;input_problem_Quantum_teleportation_3_1&#34; aria-describedby=&#34;status_problem_Quantum_teleportation_3_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_problem_Quantum_teleportation_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Quantum_teleportation_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_problem_Quantum_teleportation_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_problem_Quantum_teleportation_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_problem_Quantum_teleportation_3_1_dynamath&#34; name=&#34;input_problem_Quantum_teleportation_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; Find [mathjaxinline]U[/mathjaxinline] that [mathjaxinline]B[/mathjaxinline] needs to apply so that [mathjaxinline]B[/mathjaxinline] obtains [mathjaxinline]|{\psi }\rangle[/mathjaxinline] up to global phases for the following measurement results. &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;00&amp;#160;&amp;#160;&amp;#160;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;inputtype option-input inline&#34;&gt; &lt;select name=&#34;input_problem_Quantum_teleportation_4_1&#34; id=&#34;input_problem_Quantum_teleportation_4_1&#34; aria-describedby=&#34;status_problem_Quantum_teleportation_4_1&#34;&gt; &lt;option value=&#34;option_problem_Quantum_teleportation_4_1_dummy_default&#34;&gt;Select an option&lt;/option&gt; &lt;option value=&#34;I&#34;&gt; I&lt;/option&gt; &lt;option value=&#34;X&#34;&gt; X&lt;/option&gt; &lt;option value=&#34;Y&#34;&gt; Y&lt;/option&gt; &lt;option value=&#34;Z&#34;&gt; Z&lt;/option&gt; &lt;/select&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Quantum_teleportation_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;p class=&#34;answer&#34; id=&#34;answer_problem_Quantum_teleportation_4_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;01&amp;#160;&amp;#160;&amp;#160;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;inputtype option-input inline&#34;&gt; &lt;select name=&#34;input_problem_Quantum_teleportation_5_1&#34; id=&#34;input_problem_Quantum_teleportation_5_1&#34; aria-describedby=&#34;status_problem_Quantum_teleportation_5_1&#34;&gt; &lt;option value=&#34;option_problem_Quantum_teleportation_5_1_dummy_default&#34;&gt;Select an option&lt;/option&gt; &lt;option value=&#34;I&#34;&gt; I&lt;/option&gt; &lt;option value=&#34;X&#34;&gt; X&lt;/option&gt; &lt;option value=&#34;Y&#34;&gt; Y&lt;/option&gt; &lt;option value=&#34;Z&#34;&gt; Z&lt;/option&gt; &lt;/select&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Quantum_teleportation_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;p class=&#34;answer&#34; id=&#34;answer_problem_Quantum_teleportation_5_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;10&amp;#160;&amp;#160;&amp;#160;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;inputtype option-input inline&#34;&gt; &lt;select name=&#34;input_problem_Quantum_teleportation_6_1&#34; id=&#34;input_problem_Quantum_teleportation_6_1&#34; aria-describedby=&#34;status_problem_Quantum_teleportation_6_1&#34;&gt; &lt;option value=&#34;option_problem_Quantum_teleportation_6_1_dummy_default&#34;&gt;Select an option&lt;/option&gt; &lt;option value=&#34;I&#34;&gt; I&lt;/option&gt; &lt;option value=&#34;X&#34;&gt; X&lt;/option&gt; &lt;option value=&#34;Y&#34;&gt; Y&lt;/option&gt; &lt;option value=&#34;Z&#34;&gt; Z&lt;/option&gt; &lt;/select&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Quantum_teleportation_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;p class=&#34;answer&#34; id=&#34;answer_problem_Quantum_teleportation_6_1&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;11&amp;#160;&amp;#160;&amp;#160;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;inputtype option-input inline&#34;&gt; 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Superdense_Coding__1_of_3_" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="vertical"> <h2 class="hd hd-2 unit-title">Superdense Coding (1 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__1_of_3_"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__1_of_3_" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="problem"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_problem_Superdense_Coding__1_of_3_" class="problems-wrapper" role="group" aria-labelledby="problem_Superdense_Coding__1_of_3_-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__1_of_3_" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__1_of_3_/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="4" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;problem_Superdense_Coding__1_of_3_-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__1_of_3_-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Superdense Coding (1 of 3) &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__1_of_3_-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Superdense coding is the protocol where [mathjaxinline]A[/mathjaxinline] can send 2 bits of information by only sending 1 qubit to [mathjaxinline]B[/mathjaxinline] given a preshared entanglement among [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline]. We will see how this protocol works. In this problem, we consider a slightly modified protocol from the one presented in the lecture where we use a different Bell state. &lt;/p&gt; &lt;p&gt; Let [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] share [mathjaxinline]|{\Psi ^-}\rangle[/mathjaxinline]. What is the state when [mathjaxinline]A[/mathjaxinline] applies [mathjaxinline]I,X,Z,Y[/mathjaxinline]. Answer in the ket notation with [mathjaxinline]\{ |{00}\rangle ,|{01}\rangle ,|{10}\rangle ,|{11}\rangle \}[/mathjaxinline] basis. Take the global phase so that the coefficients are real. &lt;/p&gt; &lt;ul class=&#34;itemize&#34;&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]I[/mathjaxinline]&lt;br/&gt;&lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Superdense_Coding__1_of_3__2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_problem_Superdense_Coding__1_of_3__2_1&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__2_1&#34; aria-describedby=&#34;status_problem_Superdense_Coding__1_of_3__2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_problem_Superdense_Coding__1_of_3__2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Superdense_Coding__1_of_3__2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_problem_Superdense_Coding__1_of_3__2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_problem_Superdense_Coding__1_of_3__2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__2_1_dynamath&#34; name=&#34;input_problem_Superdense_Coding__1_of_3__2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]X[/mathjaxinline]&lt;br/&gt;&lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Superdense_Coding__1_of_3__3_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_problem_Superdense_Coding__1_of_3__3_1&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__3_1&#34; aria-describedby=&#34;status_problem_Superdense_Coding__1_of_3__3_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_problem_Superdense_Coding__1_of_3__3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Superdense_Coding__1_of_3__3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_problem_Superdense_Coding__1_of_3__3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_problem_Superdense_Coding__1_of_3__3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__3_1_dynamath&#34; name=&#34;input_problem_Superdense_Coding__1_of_3__3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]Z[/mathjaxinline]&lt;br/&gt;&lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Superdense_Coding__1_of_3__4_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_problem_Superdense_Coding__1_of_3__4_1&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__4_1&#34; aria-describedby=&#34;status_problem_Superdense_Coding__1_of_3__4_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_problem_Superdense_Coding__1_of_3__4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Superdense_Coding__1_of_3__4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_problem_Superdense_Coding__1_of_3__4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_problem_Superdense_Coding__1_of_3__4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__4_1_dynamath&#34; name=&#34;input_problem_Superdense_Coding__1_of_3__4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/li&gt; &lt;li&gt; &lt;p&gt; [mathjaxinline]Y[/mathjaxinline]&lt;br/&gt;&lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Superdense_Coding__1_of_3__5_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_problem_Superdense_Coding__1_of_3__5_1&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__5_1&#34; aria-describedby=&#34;status_problem_Superdense_Coding__1_of_3__5_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_problem_Superdense_Coding__1_of_3__5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Superdense_Coding__1_of_3__5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_problem_Superdense_Coding__1_of_3__5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_problem_Superdense_Coding__1_of_3__5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_problem_Superdense_Coding__1_of_3__5_1_dynamath&#34; 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Superdense_Coding__2_of_3_" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="vertical"> <h2 class="hd hd-2 unit-title">Superdense Coding (2 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__2_of_3_"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__2_of_3_" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="problem"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_problem_Superdense_Coding__2_of_3_" class="problems-wrapper" role="group" aria-labelledby="problem_Superdense_Coding__2_of_3_-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__2_of_3_" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__2_of_3_/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;problem_Superdense_Coding__2_of_3_-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__2_of_3_-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Superdense Coding (2 of 3) &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__2_of_3_-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose [mathjaxinline]A[/mathjaxinline] would like to send 2 bits [mathjaxinline]xy[/mathjaxinline] by the superdense coding protocol. [mathjaxinline]A[/mathjaxinline] applies some unitaries to her part of the Bell pair, sends her qubit to [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]B[/mathjaxinline] makes the Bell measurement. Suppose [mathjaxinline]B[/mathjaxinline] carries out the Bell measurement that is described in the previous teleportation problem to obtain the 2 bits of information. Complete the following quantum circuit for [mathjaxinline]A[/mathjaxinline] where [mathjaxinline]A[/mathjaxinline] encodes the 2 bits of information so that [mathjaxinline]B[/mathjaxinline] obtains the same 2 bits that [mathjaxinline]A[/mathjaxinline] would like to send. Note that the single wire corresponds to the one of the qubits of the preshared entangled state which is obtained by [mathjaxinline]A[/mathjaxinline], and double wires refer to the classical control, i.e. if [mathjaxinline]x[/mathjaxinline] is 1, it triggers the unitary in the box below etc. &lt;/p&gt; &lt;span&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Superdense_Coding__2_of_3__2_1&#34; class=&#34;capa_inputtype&#34;&gt; &lt;div class=&#34;drag_and_drop_problem_div&#34; id=&#34;drag_and_drop_div_problem_Superdense_Coding__2_of_3__2_1&#34; data-plain-id=&#34;problem_Superdense_Coding__2_of_3__2_1&#34;&gt; &lt;/div&gt; &lt;div class=&#34;drag_and_drop_problem_json&#34; id=&#34;drag_and_drop_json_problem_Superdense_Coding__2_of_3__2_1&#34; style=&#34;display:none;&#34;&gt;{&#34;targets&#34;: [{&#34;w&#34;: &#34;118&#34;, &#34;id&#34;: &#34;1&#34;, &#34;y&#34;: &#34;103&#34;, &#34;h&#34;: &#34;84&#34;, &#34;x&#34;: &#34;49&#34;}, {&#34;w&#34;: &#34;117&#34;, &#34;id&#34;: &#34;3&#34;, &#34;y&#34;: &#34;103&#34;, &#34;h&#34;: &#34;84&#34;, &#34;x&#34;: &#34;286&#34;}, {&#34;w&#34;: &#34;118&#34;, &#34;id&#34;: &#34;2&#34;, &#34;y&#34;: &#34;103&#34;, &#34;h&#34;: &#34;84&#34;, &#34;x&#34;: &#34;522&#34;}], &#34;draggables&#34;: [{&#34;id&#34;: &#34;I&#34;, &#34;can_reuse&#34;: &#34;true&#34;, &#34;label&#34;: &#34;&#34;, &#34;target_fields&#34;: [], &#34;icon&#34;: &#34;/assets/courseware/v1/cb88a3e04332b5966927af86e2aeae1e/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_U2P3A_sd_enc_circ_U2P3A_sd_enc_circ_dnd_label1.png&#34;}, {&#34;id&#34;: &#34;X&#34;, &#34;can_reuse&#34;: &#34;true&#34;, &#34;label&#34;: &#34;&#34;, &#34;target_fields&#34;: [], &#34;icon&#34;: &#34;/assets/courseware/v1/2f41a336954992b8f1f7f44664dc98c0/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_U2P3A_sd_enc_circ_U2P3A_sd_enc_circ_dnd_label2.png&#34;}, {&#34;id&#34;: &#34;Z&#34;, &#34;can_reuse&#34;: &#34;true&#34;, &#34;label&#34;: &#34;&#34;, &#34;target_fields&#34;: [], &#34;icon&#34;: &#34;/assets/courseware/v1/de7ed3d640426e647299aa3f4acab781/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_U2P3A_sd_enc_circ_U2P3A_sd_enc_circ_dnd_label3.png&#34;}, {&#34;id&#34;: &#34;Y&#34;, &#34;can_reuse&#34;: &#34;true&#34;, &#34;label&#34;: &#34;&#34;, &#34;target_fields&#34;: [], &#34;icon&#34;: &#34;/assets/courseware/v1/c439d62e1db42e12a9321f9f5d0003ca/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_U2P3A_sd_enc_circ_U2P3A_sd_enc_circ_dnd_label4.png&#34;}, {&#34;id&#34;: &#34;H&#34;, &#34;can_reuse&#34;: &#34;true&#34;, &#34;label&#34;: &#34;&#34;, &#34;target_fields&#34;: [], &#34;icon&#34;: &#34;/assets/courseware/v1/27744823709fdcfdb4521056eb908e63/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_U2P3A_sd_enc_circ_U2P3A_sd_enc_circ_dnd_label5.png&#34;}], &#34;one_per_target&#34;: &#34;true&#34;, &#34;label_bg_color&#34;: &#34;rgb(222, 139, 238)&#34;, &#34;base_image&#34;: &#34;/assets/courseware/v1/71c26413927d824d3b41682889bfba03/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_U2P3A_sd_enc_circ_U2P3A_sd_enc_circ_dnd.png&#34;, &#34;target_outline&#34;: &#34;false&#34;}&lt;/div&gt; 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&lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_problem_Superdense_Coding__2_of_3__solution_1&#34;/&gt; &lt;/div&gt;&lt;/span&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_problem_Superdense_Coding__2_of_3__solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Superdense Coding (2 of 3)&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_problem_Superdense_Coding__2_of_3_&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_problem_Superdense_Coding__2_of_3_&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;problem_Superdense_Coding__2_of_3_-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;problem_Superdense_Coding__2_of_3_-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;problem_Superdense_Coding__2_of_3_-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.370.2x+1T2018+type@html+block@html_site_search_box2x"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@html+block@html_site_search_box2x" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="html"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <span><a href="/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.370.2x+1T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript"> var add_site_search = function(){ course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', ''); jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', ''); if (typeof String.prototype.startsWith != 'function') { // see below for better implementation! String.prototype.startsWith = function (str){ return this.indexOf(str) === 0; }; } if(typeof(String.prototype.trim) === "undefined") { String.prototype.trim = function() { return String(this).replace(/^\s+|\s+$/g, ''); }; } var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var amp = String.fromCharCode(38); var rlb = rb + lb; var mke = function(x){ return lb + x + rb; } var search_module_url = ""; var get_search_module_ficus = function(){ var cid = $('div.xblock').data('course-id'); if (cid){ console.log("cid = ", cid); // search_module_url = "/courses/course-v1:MITx+8.370.2x+1T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.370.2x+1T2018/courseware/welcome/Search_this_course/"; // automatically rewritten console.log("3. search_module_url = ", search_module_url); return; } var course_root_link = $('span.nav-item-course').find('a').attr('href'); if (course_root_link){ console.log("course_root_link = ", course_root_link); search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/"); console.log("2. search_module_url = ", search_module_url); return } console.log("cannot determine search module url"); } var get_search_module = function(){ // find search this module link if (!($('div.course-index').length)){ return get_search_module_ficus(); } $('div.course-index').find('nav').find('a').each(function(){ if ($(this).text().trim().startsWith("Search this course")){ search_module_url = $(this).attr('href'); console.log("search_module_url = ", search_module_url); } }); } var go_to_search = function(){ get_search_module(); var sterm = $('#site-search-box').val(); // new_url = jump_to_url + "/Search_this_module/?q=" + sterm; new_url = search_module_url + "?q=" + sterm; console.log("sterm = ", sterm, " ; going to ", new_url); window.location.href = new_url; } if (!$('#site-search-box').length){ $("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'" + " id='site-search-box'" + rlb + "img src='" + course_static_url + "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb); } $("#site-search-box").keypress(function(event) { if (event.which == 13) { event.preventDefault(); go_to_search(); } }); // $('#site-search-box').bind("enterKey", go_to_search); var get = function(x){ return eval(x); } return {'course_static_url': course_static_url, 'jump_to_url': jump_to_url, 'go_to_search': go_to_search, 'get_search_module': get_search_module, 'get_search_module_ficus': get_search_module_ficus, 'get': get, } } var the_site_search = add_site_search(); var add_fix_transcript = function(){ if ($('div.wrap-instructor-info').length==0){ return; } $('div.xblock-student_view-video').each(function(key, vblock_e){ var vblock = $(vblock_e); var vuid = vblock.data('usage-id').split('@'); var vid; if (vuid.length==1){ vuid = vblock.data('usage-id').split(';_') vid = vuid[5]; }else{ vid = vuid[2]; } var mfnpre = vid.split("_video",1)[0]; var mfnid = mfnpre; // no periods mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var mke = function(x){ return lb + x + rb; } var ftid = "fix_transcript_" + mfnid; if (!$('#' + ftid).length){ var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb; html += "contribute transcript fix" + mke("/a") + mke("/span"); console.log("html = ", html); vblock.after(html) } $('#' + ftid).click(function(){ var cst = $('ol.subtitles').find('li.current'); var cindex = Number(cst.data('index')); var gurl; if (mfnpre.endsWith('_cq_sol')){ gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/"; }else{ gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/"; } gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1); console.log("going to ", gurl); window.open(gurl, "MITx 8.370x subtitle source"); }); }); } try{ add_fix_transcript(); } catch(err){ console.log(err); } try{ var rb = String.fromCharCode(62); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000); } catch(err){ console.log(err); } </script></span> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@vertical+block@Superdense_Coding__3_of_3_" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="vertical"> <h2 class="hd hd-2 unit-title">Superdense Coding (3 of 3)</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__3_of_3_"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__3_of_3_" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="problem"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_problem_Superdense_Coding__3_of_3_" class="problems-wrapper" role="group" aria-labelledby="problem_Superdense_Coding__3_of_3_-problem-title" data-problem-id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__3_of_3_" data-url="/courses/course-v1:MITx+8.370.2x+1T2018/xblock/block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__3_of_3_/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;problem_Superdense_Coding__3_of_3_-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__3_of_3_-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Superdense Coding (3 of 3) &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.370.2x+1T2018+type@problem+block@problem_Superdense_Coding__3_of_3_-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] share [mathjaxinline]|{\psi }\rangle =\frac{1}{\sqrt{2}}\left(\sqrt{1-\epsilon }|{01}\rangle -\sqrt{1+\epsilon }|{10}\rangle \right)[/mathjaxinline] instead of the perfect [mathjaxinline]|{\Psi ^-}\rangle[/mathjaxinline]. Suppose they follow the same protocol as above. Find the probability that [mathjaxinline]A[/mathjaxinline] fails to send her 2 bits of information to [mathjaxinline]B[/mathjaxinline] up to the leading order of [mathjaxinline]\epsilon[/mathjaxinline]. Input &lt;tt class=&#34;ttfamily&#34;&gt;E&lt;/tt&gt; for [mathjaxinline]\epsilon[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_problem_Superdense_Coding__3_of_3__2_1&#34; class=&#34;text-input-dynamath capa_inputtype textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data &#34; data-preprocessor=&#34;MathjaxPreprocessorForQM&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js&#34;/&gt; &lt;div class=&#34;unanswered &#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_problem_Superdense_Coding__3_of_3__2_1&#34; id=&#34;input_problem_Superdense_Coding__3_of_3__2_1&#34; aria-describedby=&#34;status_problem_Superdense_Coding__3_of_3__2_1&#34; value=&#34;&#34; class=&#34;math&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_problem_Superdense_Coding__3_of_3__2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_problem_Superdense_Coding__3_of_3__2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_problem_Superdense_Coding__3_of_3__2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_problem_Superdense_Coding__3_of_3__2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_problem_Superdense_Coding__3_of_3__2_1_dynamath&#34; name=&#34;input_problem_Superdense_Coding__3_of_3__2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_problem_Superdense_Coding__3_of_3__solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Superdense Coding (3 of 3)&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_problem_Superdense_Coding__3_of_3_&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_problem_Superdense_Coding__3_of_3_&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;problem_Superdense_Coding__3_of_3_-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;problem_Superdense_Coding__3_of_3_-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;problem_Superdense_Coding__3_of_3_-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.370.2x+1T2018+type@html+block@html_site_search_box1xx"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.2x+1T2018+type@html+block@html_site_search_box1xx" data-request-token="3d4f5d881edc11f097b212192c274abf" data-runtime-version="1" data-graded="True" data-course-id="course-v1:MITx+8.370.2x+1T2018" data-runtime-class="LmsRuntime" data-block-type="html"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <span><a href="/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.370.2x+1T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript"> var add_site_search = function(){ course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', ''); jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', ''); if (typeof String.prototype.startsWith != 'function') { // see below for better implementation! String.prototype.startsWith = function (str){ return this.indexOf(str) === 0; }; } if(typeof(String.prototype.trim) === "undefined") { String.prototype.trim = function() { return String(this).replace(/^\s+|\s+$/g, ''); }; } var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var amp = String.fromCharCode(38); var rlb = rb + lb; var mke = function(x){ return lb + x + rb; } var search_module_url = ""; var get_search_module_ficus = function(){ var cid = $('div.xblock').data('course-id'); if (cid){ console.log("cid = ", cid); // search_module_url = "/courses/course-v1:MITx+8.370.2x+1T2018/" + cid + "/courseware/welcome/Search_this_course/"; search_module_url = "/courses/course-v1:MITx+8.370.2x+1T2018/courseware/welcome/Search_this_course/"; // automatically rewritten console.log("3. search_module_url = ", search_module_url); return; } var course_root_link = $('span.nav-item-course').find('a').attr('href'); if (course_root_link){ console.log("course_root_link = ", course_root_link); search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/"); console.log("2. search_module_url = ", search_module_url); return } console.log("cannot determine search module url"); } var get_search_module = function(){ // find search this module link if (!($('div.course-index').length)){ return get_search_module_ficus(); } $('div.course-index').find('nav').find('a').each(function(){ if ($(this).text().trim().startsWith("Search this course")){ search_module_url = $(this).attr('href'); console.log("search_module_url = ", search_module_url); } }); } var go_to_search = function(){ get_search_module(); var sterm = $('#site-search-box').val(); // new_url = jump_to_url + "/Search_this_module/?q=" + sterm; new_url = search_module_url + "?q=" + sterm; console.log("sterm = ", sterm, " ; going to ", new_url); window.location.href = new_url; } if (!$('#site-search-box').length){ $("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'" + " id='site-search-box'" + rlb + "img src='" + course_static_url + "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb); } $("#site-search-box").keypress(function(event) { if (event.which == 13) { event.preventDefault(); go_to_search(); } }); // $('#site-search-box').bind("enterKey", go_to_search); var get = function(x){ return eval(x); } return {'course_static_url': course_static_url, 'jump_to_url': jump_to_url, 'go_to_search': go_to_search, 'get_search_module': get_search_module, 'get_search_module_ficus': get_search_module_ficus, 'get': get, } } var the_site_search = add_site_search(); var add_fix_transcript = function(){ if ($('div.wrap-instructor-info').length==0){ return; } $('div.xblock-student_view-video').each(function(key, vblock_e){ var vblock = $(vblock_e); var vuid = vblock.data('usage-id').split('@'); var vid; if (vuid.length==1){ vuid = vblock.data('usage-id').split(';_') vid = vuid[5]; }else{ vid = vuid[2]; } var mfnpre = vid.split("_video",1)[0]; var mfnid = mfnpre; // no periods mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename var lb = String.fromCharCode(60); var rb = String.fromCharCode(62); var mke = function(x){ return lb + x + rb; } var ftid = "fix_transcript_" + mfnid; if (!$('#' + ftid).length){ var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb; html += "contribute transcript fix" + mke("/a") + mke("/span"); console.log("html = ", html); vblock.after(html) } $('#' + ftid).click(function(){ var cst = $('ol.subtitles').find('li.current'); var cindex = Number(cst.data('index')); var gurl; if (mfnpre.endsWith('_cq_sol')){ gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/"; }else{ gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/"; } gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1); console.log("going to ", gurl); window.open(gurl, "MITx 8.370x subtitle source"); }); }); } try{ add_fix_transcript(); } catch(err){ console.log(err); } try{ var rb = String.fromCharCode(62); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000); setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000); } catch(err){ console.log(err); } </script></span> </div> </div> </div> </div>