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<h2 class="hd hd-2 unit-title">CQ: Measurement outcome probabilities</h2>
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Measurement outcome probabilities
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This is a concept question, provided to double-check your understanding of the previous video clip. </p>
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For which of the following states is the probability of obtaining the outcome &#8220;0" equal to the probability of obtaining the outcome &#8220;2"? </p>
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<div class="video-player">
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Your browser does not support this video format. Try using a different browser.
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$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
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var rb = String.fromCharCode(62);
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Tensor product of X and Y
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
In lecture, Prof. Shor explained how to compute the <a href="https://en.wikipedia.org/wiki/Tensor_product" target="_blank">tensor product</a> of matrices. He also (in an earlier lecture) defined the Pauli matrices, which are also commonly referred to using [mathjaxinline]X[/mathjaxinline], [mathjaxinline]Y[/mathjaxinline], and [mathjaxinline]Z[/mathjaxinline] instead of [mathjaxinline]\sigma _ x[/mathjaxinline], [mathjaxinline]\sigma _ y[/mathjaxinline], and [mathjaxinline]\sigma _ z[/mathjaxinline], as we do so below. </p>
<p>
Recall that </p>
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[mathjaxinline]\displaystyle X[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left[ \begin{array}{cc}{0}&amp; {1}\\ {1}&amp; {0}\end{array}\right][/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.2)</td>
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<tr id="a0000000006">
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<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle Y[/mathjaxinline]
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<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
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<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left[ \begin{array}{cc}{0}&amp; {-i}\\ {i}&amp; {0}\end{array}\right] \, .[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.3)</td>
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Which of the following expressions correctly gives [mathjaxinline]X\otimes Y[/mathjaxinline]? </p>
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<text>[mathjaxinline]\displaystyle \left(\begin{array}{cccc}0&amp; 0&amp; 0&amp; -i \\ 0&amp; 0&amp; i&amp; 0 \\ 0&amp; -i&amp; 0&amp; 0 \\ i&amp; 0&amp; 0&amp; 0\end{array}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(\begin{array}{cccc}0&amp; 0&amp; 0&amp; -i \\ 0&amp; 0&amp; -i&amp; 0 \\ 0&amp; i&amp; 0&amp; 0 \\ i&amp; 0&amp; 0&amp; 0\end{array}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(\begin{array}{cccc}0&amp; 0&amp; 0&amp; 1 \\ 0&amp; 0&amp; 1&amp; 0 \\ 0&amp; 1&amp; 0&amp; 0 \\ 1&amp; 0&amp; 0&amp; 0\end{array}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(\begin{array}{cccc}0&amp; 0&amp; 0&amp; -1 \\ 0&amp; 0&amp; 1&amp; 0 \\ 0&amp; -1&amp; 0&amp; 0 \\ 1&amp; 0&amp; 0&amp; 0\end{array}\right)[/mathjaxinline]</text>
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Matrix description of quantum circuit
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Consider the following quantum circuit [mathjaxinline]C[/mathjaxinline], composed of two NOT gates and one CNOT gate: </p>
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Complete the following expression for the unitary matrix describing the transformation performed by [mathjaxinline]C[/mathjaxinline]: </p>
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$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
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console.log("search_module_url = ", search_module_url);
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}
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$("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();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
})();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
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vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
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vid = vuid[2];
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mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + 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')){
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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);
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catch(err){
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<h2 class="hd hd-2 unit-title">Multi-qubit tensor product operations: cautionary note</h2>
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jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', '');
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String.prototype.startsWith = function (str){
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{
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var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
go_to_search = function(){
// find search this module link
search_module_url = "";
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
var 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();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
})();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
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var vuid = vblock.data('usage-id').split('@');
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vid = vuid[5];
}else{
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}
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;
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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)
}
$('#' + 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);
}
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</div>
</div>
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