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<h2 class="hd hd-2 unit-title">Density matrices review material: overview</h2>
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<p><big class="xlarge">Density matrices review material: overview</big></p><p>
This section includes review material from Prof. Peter Shor's 8.370 lectures on </p><ul class="itemize"><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2a" target="_blank">what noisy quantum states are</a>, </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2b" target="_blank">the indistinguishability of density matrix ensemble unravelings</a>, </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2c" target="_blank">measurements and density matrices</a>, </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2d" target="_blank">the action of unitary transforms on density matrices</a>, </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2e" target="_blank">the effect of measurement on an ensemble of states</a>, </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2f" target="_blank">density matrix properties</a>, </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2h" target="_blank">density matrices and the partial trace</a>, and </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/U3L2m" target="_blank">the Bloch sphere representation of single qubit density matrices</a>. </p></li></ul><p>
Also available are two of Prof. Isaac Chuang's tablet videos from 8.371, about </p><ul class="itemize"><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/Q1L1a" target="_blank">quantum mechanics and quantum circuits</a>, and </p></li><li><p><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/Q1L2a" target="_blank">unravelings of density matrices</a>. </p></li></ul><p>
Additional review material about quantum states and quantum measurement can be found in <a href="https://courses.edx.org/courses/course-v1:MITx+8.370.1x+1T2018/course/" target="_blank">the 8.370.1x online course</a>, and in <a href="https://www.edx.org/course/mastering-quantum-mechanics-part-1-wave-mitx-8-05-1x-0" target="_blank">8.05x</a>. </p>
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<h2 class="hd hd-2 unit-title">CQ: Density matrix unravelings</h2>
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Density matrix unravelings
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<div class="problem">
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<p>
This is a concept question, provided to check your understanding of the previous video clip. </p>
<p>
Which of the following statements about <a href="https://en.wikipedia.org/wiki/Density_matrix" target="_blank">density matrices</a>, and their unravelings into ensembles of pure states, are correct? Check all that apply </p>
<p>
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<text>Every density matrix has an infinite number of unravelings into different probabilistic ensembles of pure states.</text>
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<input type="checkbox" name="input_u3_2_cq_denmat_unravelings_2_1[]" id="input_u3_2_cq_denmat_unravelings_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="u3_2_cq_denmat_unravelings_2_1-choice_1-label" for="input_u3_2_cq_denmat_unravelings_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_u3_2_cq_denmat_unravelings_2_1">
<text>Only mixed state density matrices have an infinite number of unravelings into different probabilistic ensembles of pure states.</text>
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<text>Two different unravelings of a density matrix can be distinguished by an appropriate experiment, in principle.</text>
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<text>Different unravelings of a density matrix can never be distinguished by an appropriate experiment.</text>
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<text>Pure states can be represented by density matrices.</text>
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<div class="field">
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<text>Density matrices only represent mixed states.</text>
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<span id="answer_u3_2_cq_denmat_unravelings_2_1"/>
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<div class="video-player">
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<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
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new_url = search_module_url + "?q=" + sterm;
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}
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var get = function(x){
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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)
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$('#' + ftid).click(function(){
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var cindex = Number(cst.data('index'));
var gurl;
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<h2 class="hd hd-2 unit-title">CQ: Measurement and density matrices</h2>
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Measurement and density matrices
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This is a concept question, provided to check your understanding of the previous video clip. </p>
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Which of the following statements about <ref/>quantum measurement and density matrices are correct? Check all that apply: </p>
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<text>After a quantum measurement on a density matrix state, the post-measurement state is always a pure state.</text>
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<text>The post-measurement state is always a mixed state.</text>
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<text>After quantum measurement on a pure state, the post-measurement state is always a mixed state.</text>
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<text>After quantum measurement on a pure state, if the measurement result is known, the post-measurement state is always a pure state.</text>
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<text>Measurement on a density matrix represented by a specific ensemble of pure states, always simply projects into one of the ensemble of states.</text>
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<text>Measurement on a density matrix state sometimes results in a pure state, but may also result in a mixed state.</text>
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if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
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var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
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var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + 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();
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console.log(err);
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try{
var rb = String.fromCharCode(62);
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setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
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catch(err){
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class="video closed"
data-metadata='{"publishCompletionUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2f/handler/publish_completion", "duration": 0.0, "transcriptLanguage": "en", "ytTestTimeout": 1500, "prioritizeHls": false, "generalSpeed": 1.0, "sources": [], "captionDataDir": null, "autoAdvance": false, "poster": null, "autohideHtml5": false, "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2f/handler/transcript/translation/__lang__", "recordedYoutubeIsAvailable": true, "completionEnabled": false, "end": 0.0, "ytMetadataEndpoint": "", "completionPercentage": 0.95, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "start": 0.0, "streams": "1.00:7z0FrtuNLJE", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2f/handler/transcript/available_translations", "autoplay": false, "showCaptions": "true", "transcriptLanguages": {"en": "English"}, "saveStateUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2f/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="U3L2f"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
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</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box6">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="3016ff5c21a911f0975116ffcd2b1d4d" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box6" data-runtime-class="LmsRuntime" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
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<span><a href="/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.1x+2T2018/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+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/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>
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<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-graded="True" data-init="VerticalStudentView" data-request-token="3016ff5c21a911f0975116ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Density_matrices_and_the_partial_trace_-_example" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Density matrices and the partial trace - example</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2h">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="3016ff5c21a911f0975116ffcd2b1d4d" data-runtime-version="1" data-block-type="video" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2h" data-runtime-class="LmsRuntime" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Density matrices and the partial trace - example</h3>
<div
id="video_U3L2h"
class="video closed"
data-metadata='{"publishCompletionUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2h/handler/publish_completion", "duration": 0.0, "transcriptLanguage": "en", "ytTestTimeout": 1500, "prioritizeHls": false, "generalSpeed": 1.0, "sources": [], "captionDataDir": null, "autoAdvance": false, "poster": null, "autohideHtml5": false, "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2h/handler/transcript/translation/__lang__", "recordedYoutubeIsAvailable": true, "completionEnabled": false, "end": 0.0, "ytMetadataEndpoint": "", "completionPercentage": 0.95, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "start": 0.0, "streams": "1.00:Oo4bGX1GUes", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2h/handler/transcript/available_translations", "autoplay": false, "showCaptions": "true", "transcriptLanguages": {"en": "English"}, "saveStateUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@U3L2h/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="U3L2h"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box8">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="3016ff5c21a911f0975116ffcd2b1d4d" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box8" data-runtime-class="LmsRuntime" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<span><a href="/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.1x+2T2018/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+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten
console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
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console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('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>
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The partial trace
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This is a concept question, provided to check your understanding of the previous video clip. </p>
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Which of the following statements about the <a href="https://en.wikipedia.org/wiki/Partial_trace" target="_blank">partial trace</a> operation are correct? Check all that apply: </p>
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<text>The partial trace operation represents what happens when part of a system is discarded, or forgotten about.</text>
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<text>The patial trace operation represents what happens when part of a system is measured, and the measurement result is known.</text>
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<text>The partial trace operation on [mathjaxinline]\rho _{AB}[/mathjaxinline] can be implemented by a unitary transform on the [mathjaxinline]AB[/mathjaxinline] system.</text>
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<text>The result after a partial trace depends on the basis in which the trace is computed.</text>
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<text>The result after a partial trace is independent of the basis in which the trace is computed.</text>
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<h2 class="hd hd-2 unit-title">Quantum mechanics and quantum circuits</h2>
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<div class="video-player-pre"></div>
<div class="video-player">
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<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
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<div class="closed-captions"></div>
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<script type="json/xblock-args" class="xblock-json-init-args">
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// 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, '');
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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(){
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console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
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var get_search_module = function(){
// find search this module link
if (!($('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")){
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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){
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}
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<h2 class="hd hd-2 unit-title">Density matrices</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@Q1L2a">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="3016ff5c21a911f0975116ffcd2b1d4d" data-runtime-version="1" data-block-type="video" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@Q1L2a" data-runtime-class="LmsRuntime" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
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<h3 class="hd hd-2">Density matrices</h3>
<div
id="video_Q1L2a"
class="video closed"
data-metadata='{"publishCompletionUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@Q1L2a/handler/publish_completion", "duration": 0.0, "transcriptLanguage": "en", "ytTestTimeout": 1500, "prioritizeHls": false, "generalSpeed": 1.0, "sources": [], "captionDataDir": null, "autoAdvance": false, "poster": null, "autohideHtml5": false, "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@Q1L2a/handler/transcript/translation/__lang__", "recordedYoutubeIsAvailable": true, "completionEnabled": false, "end": 0.0, "ytMetadataEndpoint": "", "completionPercentage": 0.95, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateEnabled": false, "start": 0.0, "streams": "1.00:bVC2vjpqH9M", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@Q1L2a/handler/transcript/available_translations", "autoplay": false, "showCaptions": "true", "transcriptLanguages": {"en": "English"}, "saveStateUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@Q1L2a/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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data-autoadvance-enabled="False"
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tabindex="-1"
>
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<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="Q1L2a"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
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<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
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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+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/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>
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