<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@vertical+block@Communication_complexity_I" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Communication complexity I</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_I">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_I" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_problem_Communication_complexity_I" class="problems-wrapper" role="group"
aria-labelledby="problem_Communication_complexity_I-problem-title"
data-problem-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_I" data-url="/courses/course-v1:MITx+8.370.3x+1T2018/xblock/block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_I/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="problem_Communication_complexity_I-problem-title" aria-describedby="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_I-problem-progress" tabindex="-1">
Communication complexity I
</h3>
<div class="problem-progress" id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_I-problem-progress"></div>
<div class="problem">
<div>
<p>
In the scenario of distributed computing, multiple parties communicate with each other trying to achieve some computational task. In such a scenario, it is natural to ask how much communication between the parties is required to accomplish the task. This is the idea of communication complexity. </p>
<p>
Let us consider the case where there are only two parties, Alice and Bob. Suppose Alice has an [mathjaxinline]n[/mathjaxinline]-bit string [mathjaxinline]x[/mathjaxinline] and Bob has an [mathjaxinline]n[/mathjaxinline]-bit string [mathjaxinline]y[/mathjaxinline]. Their task is to compute [mathjaxinline]f(x,y)[/mathjaxinline] where [mathjaxinline]f[/mathjaxinline] is some function [mathjaxinline]f: \{ 0,1\} ^ n \otimes \{ 0,1\} ^ n \rightarrow \{ 0,1\}[/mathjaxinline]. The question is, given some specific [mathjaxinline]f[/mathjaxinline], how much communication is required between Alice and Bob to compute [mathjaxinline]f(x,y)[/mathjaxinline]? In this problem, we will look at how quantum mechanics helps to reduce the communication complexity for some specific choice of [mathjaxinline]f[/mathjaxinline]. </p>
<p>
[mathjaxinline]f[/mathjaxinline] we choose here is the function which we call [mathjaxinline]EQ'[/mathjaxinline]. It is the function that judges if two strings are equivalent or not. Namely, [mathjaxinline]EQ'(x,y) = 1[/mathjaxinline] if [mathjaxinline]x=y[/mathjaxinline] and [mathjaxinline]EQ'(x,y)=0[/mathjaxinline] if [mathjaxinline]x\neq y[/mathjaxinline]. Here, we make an important assumption, or "promise", on [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. The promise is that it will always be the case that the Hamming distance (number of bits where two strings are different) [mathjaxinline]\Delta (x,y)=0[/mathjaxinline] or [mathjaxinline]n/2[/mathjaxinline]. It is known that [mathjaxinline]\Omega (n)[/mathjaxinline] of communication is required if they are allowed to exchange classical information. </p>
<p>
What if they are instead allowed to send quantum states? </p>
<p>
How many qubits are required to be sent to solve the problem? </p>
<p>
Through the three parts of this problem, we will see that Deutch-Josza algorithm allows the complexity to be reduced in quantum case. </p>
<ul class="itemize">
<li>
<p>
Let us define [mathjaxinline]X_ i=x_ i \oplus y_ i[/mathjaxinline] where [mathjaxinline]x_ i[/mathjaxinline],[mathjaxinline]y_ i[/mathjaxinline] are [mathjaxinline]i[/mathjaxinline]th bit of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. Let [mathjaxinline]|X|[/mathjaxinline] be the number of [mathjaxinline]i[/mathjaxinline]'s that gives [mathjaxinline]X_ i = 1[/mathjaxinline]. What is [mathjaxinline]|X|[/mathjaxinline] in the following two cases? </p>
<ul class="itemize">
<li>
<p>
When [mathjaxinline]EQ'(x,y)=0[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_problem_Communication_complexity_I_2_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Communication_complexity_I_2_1" id="input_problem_Communication_complexity_I_2_1" aria-describedby="status_problem_Communication_complexity_I_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Communication_complexity_I_2_1"/>
<span class="status unanswered" id="status_problem_Communication_complexity_I_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Communication_complexity_I_2_1" class="answer"/>
<div id="display_problem_Communication_complexity_I_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Communication_complexity_I_2_1_dynamath" name="input_problem_Communication_complexity_I_2_1_dynamath"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
When [mathjaxinline]EQ'(x,y)=1[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_problem_Communication_complexity_I_3_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Communication_complexity_I_3_1" id="input_problem_Communication_complexity_I_3_1" aria-describedby="status_problem_Communication_complexity_I_3_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Communication_complexity_I_3_1"/>
<span class="status unanswered" id="status_problem_Communication_complexity_I_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Communication_complexity_I_3_1" class="answer"/>
<div id="display_problem_Communication_complexity_I_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Communication_complexity_I_3_1_dynamath" name="input_problem_Communication_complexity_I_3_1_dynamath"/>
</div>
</div></div>
</p>
</li>
</ul>
</li>
</ul>
<p>
<div class="solution-span">
<span id="solution_problem_Communication_complexity_I_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Communication complexity I" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_problem_Communication_complexity_I" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_problem_Communication_complexity_I">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_I-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_I-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_I-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@vertical+block@Communication_complexity_II" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Communication complexity II</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_II">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_II" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_problem_Communication_complexity_II" class="problems-wrapper" role="group"
aria-labelledby="problem_Communication_complexity_II-problem-title"
data-problem-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_II" data-url="/courses/course-v1:MITx+8.370.3x+1T2018/xblock/block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_II/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="problem_Communication_complexity_II-problem-title" aria-describedby="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_II-problem-progress" tabindex="-1">
Communication complexity II
</h3>
<div class="problem-progress" id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_II-problem-progress"></div>
<div class="problem">
<div>
<p>
In the scenario of distributed computing, multiple parties communicate with each other trying to achieve some computational task. In such a scenario, it is natural to ask how much communication between the parties is required to accomplish the task. This is the idea of communication complexity. </p>
<p>
Let us consider the case where there are only two parties, Alice and Bob. Suppose Alice has an [mathjaxinline]n[/mathjaxinline]-bit string [mathjaxinline]x[/mathjaxinline] and Bob has an [mathjaxinline]n[/mathjaxinline]-bit string [mathjaxinline]y[/mathjaxinline]. Their task is to compute [mathjaxinline]f(x,y)[/mathjaxinline] where [mathjaxinline]f[/mathjaxinline] is some function [mathjaxinline]f: \{ 0,1\} ^ n \otimes \{ 0,1\} ^ n \rightarrow \{ 0,1\}[/mathjaxinline]. The question is, given some specific [mathjaxinline]f[/mathjaxinline], how much communication is required between Alice and Bob to compute [mathjaxinline]f(x,y)[/mathjaxinline]? In this problem, we will look at how quantum mechanics helps to reduce the communication complexity for some specific choice of [mathjaxinline]f[/mathjaxinline]. </p>
<p>
[mathjaxinline]f[/mathjaxinline] we choose here is the function which we call [mathjaxinline]EQ'[/mathjaxinline]. It is the function that judges if two strings are equivalent or not. Namely, [mathjaxinline]EQ'(x,y) = 1[/mathjaxinline] if [mathjaxinline]x=y[/mathjaxinline] and [mathjaxinline]EQ'(x,y)=0[/mathjaxinline] if [mathjaxinline]x\neq y[/mathjaxinline]. Here, we make an important assumption, or "promise", on [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. The promise is that it will always be the case that the Hamming distance (number of bits where two strings are different) [mathjaxinline]\Delta (x,y)=0[/mathjaxinline] or [mathjaxinline]n/2[/mathjaxinline]. It is known that [mathjaxinline]\Omega (n)[/mathjaxinline] of communication is required if they are allowed to exchange classical information. </p>
<p>
What if they are instead allowed to send quantum states? </p>
<p>
How many qubits are required to be sent to solve the problem? </p>
<p>
Through the three parts of this problem, we will see that Deutch-Josza algorithm allows the complexity to be reduced in quantum case. </p>
<ul class="itemize">
<li>
<p>
Suppose Alice and Bob have [mathjaxinline]U_{A,B}[/mathjaxinline] such that [mathjaxinline]U_ A|{i}\rangle |{b}\rangle =|{i}\rangle |{b\oplus x_ i}\rangle[/mathjaxinline] and [mathjaxinline]U_ B|{i}\rangle |{b}\rangle =|{i}\rangle |{b\oplus y_ i}\rangle[/mathjaxinline]. Bob starts with the state </p>
<table id="a0000000002" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |{\psi _0}\rangle = \frac{1}{\sqrt{n}}\sum _{i=1}^ n |{i}\rangle |{-}\rangle[/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>
</table>
<p>
Bob then applies [mathjaxinline]U_ B[/mathjaxinline] to [mathjaxinline]|{\psi _0}\rangle[/mathjaxinline]. If we write [mathjaxinline]U_ B|{\psi _0}\rangle =\frac{1}{\sqrt{n}}\sum _ i|{i}\rangle |{\phi _ i}\rangle[/mathjaxinline], what is [mathjaxinline]|{\phi _ i}\rangle[/mathjaxinline]? Answer using ket notation with [mathjaxinline]\{ |{0}\rangle ,|{1}\rangle \}[/mathjaxinline] basis. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_problem_Communication_complexity_II_2_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Communication_complexity_II_2_1" id="input_problem_Communication_complexity_II_2_1" aria-describedby="status_problem_Communication_complexity_II_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Communication_complexity_II_2_1"/>
<span class="status unanswered" id="status_problem_Communication_complexity_II_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Communication_complexity_II_2_1" class="answer"/>
<div id="display_problem_Communication_complexity_II_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Communication_complexity_II_2_1_dynamath" name="input_problem_Communication_complexity_II_2_1_dynamath"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
Bob sends the state to Alice and she applies [mathjaxinline]U_ A[/mathjaxinline] to it. Suppose the state after she applies [mathjaxinline]U_ A[/mathjaxinline] looks [mathjaxinline]\frac{1}{\sqrt{n}}\sum _ i |{i}\rangle |{\phi '_ i}\rangle[/mathjaxinline], what is [mathjaxinline]|{\phi '_ i}\rangle[/mathjaxinline]? Answer in terms of [mathjaxinline]X_ i[/mathjaxinline] using ket notation with [mathjaxinline]\{ |{0}\rangle ,|{1}\rangle \}[/mathjaxinline] basis. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_problem_Communication_complexity_II_3_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Communication_complexity_II_3_1" id="input_problem_Communication_complexity_II_3_1" aria-describedby="status_problem_Communication_complexity_II_3_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Communication_complexity_II_3_1"/>
<span class="status unanswered" id="status_problem_Communication_complexity_II_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Communication_complexity_II_3_1" class="answer"/>
<div id="display_problem_Communication_complexity_II_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Communication_complexity_II_3_1_dynamath" name="input_problem_Communication_complexity_II_3_1_dynamath"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
Alice applies Hadamards to all the qubits and measure all the qubits except the last one. Suppose that she obtained the result 0 for all the qubits. What is [mathjaxinline]EQ'(x,y)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_problem_Communication_complexity_II_4_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Communication_complexity_II_4_1" id="input_problem_Communication_complexity_II_4_1" aria-describedby="status_problem_Communication_complexity_II_4_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Communication_complexity_II_4_1"/>
<span class="status unanswered" id="status_problem_Communication_complexity_II_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Communication_complexity_II_4_1" class="answer"/>
<div id="display_problem_Communication_complexity_II_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Communication_complexity_II_4_1_dynamath" name="input_problem_Communication_complexity_II_4_1_dynamath"/>
</div>
</div></div>
</p>
</li>
</ul>
<p>
<div class="solution-span">
<span id="solution_problem_Communication_complexity_II_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Communication complexity II" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_problem_Communication_complexity_II" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_problem_Communication_complexity_II">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_II-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_II-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_II-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@vertical+block@Communication_complexity_III" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Communication complexity III</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_III">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_III" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_problem_Communication_complexity_III" class="problems-wrapper" role="group"
aria-labelledby="problem_Communication_complexity_III-problem-title"
data-problem-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_III" data-url="/courses/course-v1:MITx+8.370.3x+1T2018/xblock/block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_III/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="problem_Communication_complexity_III-problem-title" aria-describedby="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_III-problem-progress" tabindex="-1">
Communication complexity III
</h3>
<div class="problem-progress" id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Communication_complexity_III-problem-progress"></div>
<div class="problem">
<div>
<p>
In the scenario of distributed computing, multiple parties communicate with each other trying to achieve some computational task. In such a scenario, it is natural to ask how much communication between the parties is required to accomplish the task. This is the idea of communication complexity. </p>
<p>
Let us consider the case where there are only two parties, Alice and Bob. Suppose Alice has an [mathjaxinline]n[/mathjaxinline]-bit string [mathjaxinline]x[/mathjaxinline] and Bob has an [mathjaxinline]n[/mathjaxinline]-bit string [mathjaxinline]y[/mathjaxinline]. Their task is to compute [mathjaxinline]f(x,y)[/mathjaxinline] where [mathjaxinline]f[/mathjaxinline] is some function [mathjaxinline]f: \{ 0,1\} ^ n \otimes \{ 0,1\} ^ n \rightarrow \{ 0,1\}[/mathjaxinline]. The question is, given some specific [mathjaxinline]f[/mathjaxinline], how much communication is required between Alice and Bob to compute [mathjaxinline]f(x,y)[/mathjaxinline]? In this problem, we will look at how quantum mechanics helps to reduce the communication complexity for some specific choice of [mathjaxinline]f[/mathjaxinline]. </p>
<p>
[mathjaxinline]f[/mathjaxinline] we choose here is the function which we call [mathjaxinline]EQ'[/mathjaxinline]. It is the function that judges if two strings are equivalent or not. Namely, [mathjaxinline]EQ'(x,y) = 1[/mathjaxinline] if [mathjaxinline]x=y[/mathjaxinline] and [mathjaxinline]EQ'(x,y)=0[/mathjaxinline] if [mathjaxinline]x\neq y[/mathjaxinline]. Here, we make an important assumption, or "promise", on [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. The promise is that it will always be the case that the Hamming distance (number of bits where two strings are different) [mathjaxinline]\Delta (x,y)=0[/mathjaxinline] or [mathjaxinline]n/2[/mathjaxinline]. It is known that [mathjaxinline]\Omega (n)[/mathjaxinline] of communication is required if they are allowed to exchange classical information. </p>
<p>
What if they are instead allowed to send quantum states? </p>
<p>
How many qubits are required to be sent to solve the problem? </p>
<p>
Through the three parts of this problem, we will see that Deutch-Josza algorithm allows the complexity to be reduced in quantum case. </p>
<ul class="itemize">
<li>
<p>
What is the complexity(required number of qubits sent between two parties) of this task in a quantum setting? Check the right answer below. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_problem_Communication_complexity_III_2_1">
<fieldset aria-describedby="status_problem_Communication_complexity_III_2_1">
<div class="field">
<input type="checkbox" name="input_problem_Communication_complexity_III_2_1[]" id="input_problem_Communication_complexity_III_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="problem_Communication_complexity_III_2_1-choice_0-label" for="input_problem_Communication_complexity_III_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_problem_Communication_complexity_III_2_1">
<text>[mathjaxinline]\mathcal{O}(2^ n)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Communication_complexity_III_2_1[]" id="input_problem_Communication_complexity_III_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="problem_Communication_complexity_III_2_1-choice_1-label" for="input_problem_Communication_complexity_III_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_problem_Communication_complexity_III_2_1">
<text>[mathjaxinline]\mathcal{O}(n)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Communication_complexity_III_2_1[]" id="input_problem_Communication_complexity_III_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="problem_Communication_complexity_III_2_1-choice_2-label" for="input_problem_Communication_complexity_III_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_problem_Communication_complexity_III_2_1">
<text>[mathjaxinline]\mathcal{O}(\log _2(n))[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Communication_complexity_III_2_1[]" id="input_problem_Communication_complexity_III_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="problem_Communication_complexity_III_2_1-choice_3-label" for="input_problem_Communication_complexity_III_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_problem_Communication_complexity_III_2_1">
<text>[mathjaxinline]\mathcal{O}(\log _2\left(\log _2(n)\right))[/mathjaxinline]</text>
</label>
</div>
<span id="answer_problem_Communication_complexity_III_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_problem_Communication_complexity_III_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
</li>
</ul>
<p>
<div class="solution-span">
<span id="solution_problem_Communication_complexity_III_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Communication complexity III" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_problem_Communication_complexity_III" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_problem_Communication_complexity_III">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_III-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_III-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Communication_complexity_III-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@vertical+block@Nonlocality_experiments" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Nonlocality experiments</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Nonlocality_experiments">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.370.3x+1T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Nonlocality_experiments" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="58363990ed9211eeb4511299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_problem_Nonlocality_experiments" class="problems-wrapper" role="group"
aria-labelledby="problem_Nonlocality_experiments-problem-title"
data-problem-id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Nonlocality_experiments" data-url="/courses/course-v1:MITx+8.370.3x+1T2018/xblock/block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Nonlocality_experiments/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="4"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="problem_Nonlocality_experiments-problem-title" aria-describedby="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Nonlocality_experiments-problem-progress" tabindex="-1">
Nonlocality experiments
</h3>
<div class="problem-progress" id="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Nonlocality_experiments-problem-progress"></div>
<div class="problem">
<div>
<p>
Ideas of communication complexity can be applied to the problem of experimentally testing the "quantumness". Bell came up with an inequality, known as Bell's inequality, which must be satisfied any local hidden variable theory, and enormous amount of efforts have been devoted to experimentally show the violation of the inequality trying to show the validity of quantum mechanics. The key is to look at a correlation that appears in quantum mechanical description but not classically. </p>
<p>
In this problem, we will see an example of such a correlation, again by borrowing the help of Deutch-Josza algorithm. We will see that communication complexity argument we saw in the previous problem is helpful to see that it is a right correlation to investigate quantum nonlocality. </p>
<p>
This time, Alice and Bob CANNOT communicate, but start with sharing a quantum state, </p>
<table id="a0000000013" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000014">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{1}{\sqrt{n}}\sum _ i |{i}\rangle _ A|{i}\rangle _ B[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.8)</td>
</tr>
</table>
<p>
They receive classical bit strings [mathjaxinline]x,y\in \{ 0,1\} ^ n[/mathjaxinline] respectively with the promise [mathjaxinline]\Delta (x,y)=0[/mathjaxinline] or [mathjaxinline]\Delta (x,y)=n/2[/mathjaxinline]. They then apply [mathjaxinline]U_ A[/mathjaxinline] and [mathjaxinline]U_ B[/mathjaxinline] discussed in the last problem while appending [mathjaxinline]|{-}\rangle[/mathjaxinline] state in their own system, i.e. [mathjaxinline]\frac{1}{\sqrt{n}}(\sum (U_ A|{i}\rangle |{-}\rangle )(U_ B|{i}\rangle |{-}\rangle ))[/mathjaxinline]. Each of them apply Hadamards to all the qubits except the last one and measure them (except the last qubit) to obtain string [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p>
<ul class="itemize">
<li>
<p>
If [mathjaxinline]x=y[/mathjaxinline], what is the probability of getting string [mathjaxinline]a,b[/mathjaxinline]? Namely, what is [mathjaxinline]{\rm Prob}[a,b|\Delta (x,y)=0][/mathjaxinline]? Answer for the following two cases. </p>
<ul class="itemize">
<li>
<p>
For [mathjaxinline]a,b[/mathjaxinline] such that [mathjaxinline]a=b[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]{\rm Prob}[a,b|\Delta (x,y)=0]=[/mathjaxinline]</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_problem_Nonlocality_experiments_2_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Nonlocality_experiments_2_1" id="input_problem_Nonlocality_experiments_2_1" aria-describedby="status_problem_Nonlocality_experiments_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Nonlocality_experiments_2_1"/>
<span class="status unanswered" id="status_problem_Nonlocality_experiments_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Nonlocality_experiments_2_1" class="answer"/>
<div id="display_problem_Nonlocality_experiments_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Nonlocality_experiments_2_1_dynamath" name="input_problem_Nonlocality_experiments_2_1_dynamath"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
For [mathjaxinline]a,b[/mathjaxinline] such that [mathjaxinline]a\neq b[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]{\rm Prob}[a,b|\Delta (x,y)=0]=[/mathjaxinline]</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_problem_Nonlocality_experiments_3_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Nonlocality_experiments_3_1" id="input_problem_Nonlocality_experiments_3_1" aria-describedby="status_problem_Nonlocality_experiments_3_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Nonlocality_experiments_3_1"/>
<span class="status unanswered" id="status_problem_Nonlocality_experiments_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Nonlocality_experiments_3_1" class="answer"/>
<div id="display_problem_Nonlocality_experiments_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Nonlocality_experiments_3_1_dynamath" name="input_problem_Nonlocality_experiments_3_1_dynamath"/>
</div>
</div></div>
</p>
</li>
</ul>
</li>
<li>
<p>
Suppose [mathjaxinline]x\neq y[/mathjaxinline]. For [mathjaxinline]a,b[/mathjaxinline] such that [mathjaxinline]a=b[/mathjaxinline], what is [mathjaxinline]{\rm Prob}[a,b|\Delta (x,y)=n/2][/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_problem_Nonlocality_experiments_4_1" class="text-input-dynamath capa_inputtype textline">
<div class="text-input-dynamath_data " data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/95fe583d41f010195cf50c9f61992d94/asset-v1:MITx+8.370.3x+1T2018+type@asset+block/mathjax_preprocessor_for_QM_H.js"/>
<div class="unanswered ">
<input type="text" name="input_problem_Nonlocality_experiments_4_1" id="input_problem_Nonlocality_experiments_4_1" aria-describedby="status_problem_Nonlocality_experiments_4_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_problem_Nonlocality_experiments_4_1"/>
<span class="status unanswered" id="status_problem_Nonlocality_experiments_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_problem_Nonlocality_experiments_4_1" class="answer"/>
<div id="display_problem_Nonlocality_experiments_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_problem_Nonlocality_experiments_4_1_dynamath" name="input_problem_Nonlocality_experiments_4_1_dynamath"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
Suppose Alice and Bob could classically obtain the string [mathjaxinline]a,b[/mathjaxinline] having the the same correlation with [mathjaxinline]x,y[/mathjaxinline] discussed above. How many classical bits would be needed to be sent to compute [mathjaxinline]EQ'(x,y)[/mathjaxinline]? Choose the right one from the following. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_problem_Nonlocality_experiments_5_1">
<fieldset aria-describedby="status_problem_Nonlocality_experiments_5_1">
<div class="field">
<input type="checkbox" name="input_problem_Nonlocality_experiments_5_1[]" id="input_problem_Nonlocality_experiments_5_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="problem_Nonlocality_experiments_5_1-choice_0-label" for="input_problem_Nonlocality_experiments_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_problem_Nonlocality_experiments_5_1">
<text>[mathjaxinline]2n[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Nonlocality_experiments_5_1[]" id="input_problem_Nonlocality_experiments_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="problem_Nonlocality_experiments_5_1-choice_1-label" for="input_problem_Nonlocality_experiments_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_problem_Nonlocality_experiments_5_1">
<text>[mathjaxinline]n[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Nonlocality_experiments_5_1[]" id="input_problem_Nonlocality_experiments_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="problem_Nonlocality_experiments_5_1-choice_2-label" for="input_problem_Nonlocality_experiments_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_problem_Nonlocality_experiments_5_1">
<text>[mathjaxinline]\lceil \frac{n}{2} \rceil[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Nonlocality_experiments_5_1[]" id="input_problem_Nonlocality_experiments_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="problem_Nonlocality_experiments_5_1-choice_3-label" for="input_problem_Nonlocality_experiments_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_problem_Nonlocality_experiments_5_1">
<text>[mathjaxinline]\lceil \log _2(n) \rceil[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Nonlocality_experiments_5_1[]" id="input_problem_Nonlocality_experiments_5_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="problem_Nonlocality_experiments_5_1-choice_4-label" for="input_problem_Nonlocality_experiments_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_problem_Nonlocality_experiments_5_1">
<text>[mathjaxinline]2^ n[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_problem_Nonlocality_experiments_5_1[]" id="input_problem_Nonlocality_experiments_5_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="problem_Nonlocality_experiments_5_1-choice_5-label" for="input_problem_Nonlocality_experiments_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_problem_Nonlocality_experiments_5_1">
<text>[mathjaxinline]2^{\lceil n/2 \rceil }[/mathjaxinline]</text>
</label>
</div>
<span id="answer_problem_Nonlocality_experiments_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_problem_Nonlocality_experiments_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
</li>
</ul>
<p>
<div class="solution-span">
<span id="solution_problem_Nonlocality_experiments_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Nonlocality experiments" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_problem_Nonlocality_experiments" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_problem_Nonlocality_experiments">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Nonlocality_experiments-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Nonlocality_experiments-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="problem_Nonlocality_experiments-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
© All Rights Reserved