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<h2 class="hd hd-2 unit-title">Typical sequences</h2>
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Typical sequences
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Let [mathjaxinline]X_1, X_2, X_3, \ldots[/mathjaxinline] be an i.i.d sequence of random variables [mathjaxinline]\cal X[/mathjaxinline] with range [mathjaxinline]\{ a,b,c\}[/mathjaxinline] and probability mass function [mathjaxinline]p(a) = 0.8[/mathjaxinline], [mathjaxinline]p(b) = p(c) = 0.1[/mathjaxinline]. </p>
<p>
Calculate the entropy per symbol [mathjaxinline]H({\cal X}) = H(X_1)[/mathjaxinline]. </p>
<p>
Express your result as a decimal number, correct to within 1%. </p>
<p>
<p style="display:inline">[mathjaxinline]H({\cal X}) =[/mathjaxinline]</p>
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Typical sequences II
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<p>
The set of [mathjaxinline]\delta[/mathjaxinline]-typical sequences of length [mathjaxinline]n[/mathjaxinline], [mathjaxinline]T_{p,\delta }^ n[/mathjaxinline], consists of sequences for which </p>
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<td class="equation" style="width:80%; border:none">[mathjax]2^{-n(H(X)+\delta )} \le p(x_1,...,x_ n) \le 2^{-n(H(X)-\delta )} \, .[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.4)</td>
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<p>
In this case, a sequence is [mathjaxinline]\delta[/mathjaxinline]-typical when </p>
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<td class="equation" style="width:80%; border:none">[mathjax]2^{-n(0.922+\delta )} \le f(n,n_ a) \le 2^{-n(0.922-\delta )} \, ,[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.5)</td>
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<p>
where [mathjaxinline]n_ a[/mathjaxinline] is the number of times &#8220;a" appears in the sequence. </p>
<p>
Give an expression for the function [mathjaxinline]f(n,n_ a)[/mathjaxinline]. Enter [mathjaxinline]{\tt na}[/mathjaxinline] for [mathjaxinline]n_ a[/mathjaxinline], and remember to explicitly include multiplication using [mathjaxinline]{\tt *}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]f(n,n_ a) =[/mathjaxinline]</p>
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Let [mathjaxinline]T_{p,0.1}^{100}[/mathjaxinline] be the set of [mathjaxinline]0.1[/mathjaxinline]-typical sequences of length [mathjaxinline]100[/mathjaxinline]. Compute [mathjaxinline]{\rm Pr}\left(T_{p,0.1}^{100} \right)[/mathjaxinline]. </p>
<p>
Express your result as a decimal number, correct to within 1%. </p>
<p>
<p style="display:inline">[mathjaxinline]{\rm Pr}\left(T_{p,0.1}^{100} \right) =[/mathjaxinline]</p>
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Typical sequences III
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Compute [mathjaxinline]|T_{p,0.1}^{100}|[/mathjaxinline], the number of typical sequences, and from this compute the number of (whole) bits needed to represent all typical sequences. </p>
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<p style="display:inline">number of bits =</p>
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Compare this with the number of bits expected from the entropy. </p>
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<p style="display:inline">[mathjaxinline]\lceil n H(x_1) \rceil =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">Quantum random access codes</h2>
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Quantum random access codes
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-raccesss-problem-progress"></div>
<div class="problem">
<div>
<p>
According to Holevo's theorem, one qubit can be used to transmit no more than one bit of information. However, this limit admits an interesting degree of freedom: it it possible to encode [mathjaxinline]m[/mathjaxinline] bits in [mathjaxinline]n[/mathjaxinline] qubits, where [mathjaxinline]m&gt;n[/mathjaxinline], in such a manner that (at most) one of the bits can be retrieved with probability at least [mathjaxinline]p[/mathjaxinline], for some [mathjaxinline]p[/mathjaxinline]. We explore this construction of <em>quantum random access codes</em> in this problem. </p>
<ol class="enumerate">
<li value="1">
<p>
Let Alice have two bits [mathjaxinline]x_0[/mathjaxinline] and [mathjaxinline]x_1[/mathjaxinline] which are random and uniformly distributed. She encodes them as a single qubit state [mathjaxinline]|\psi _{x_0 x_1}\rangle[/mathjaxinline], where </p>
<table id="a0000000015" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000016">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\psi _{00}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle |0\rangle[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.12)</td>
</tr>
<tr id="a0000000017">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\psi _{01}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (|0\rangle + |1\rangle )/\sqrt{2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.13)</td>
</tr>
<tr id="a0000000018">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\psi _{11}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle |1\rangle[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.14)</td>
</tr>
<tr id="a0000000019">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\psi _{10}\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (|0\rangle - |1\rangle )/\sqrt{2} \, .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.15)</td>
</tr>
</table>
<p>
Bob receives this qubit over the quantum channel, chooses [mathjaxinline]k=0[/mathjaxinline] or [mathjaxinline]k=1[/mathjaxinline] (randomly, with equal probability), and performs a projective measurement [mathjaxinline]M_{k}[/mathjaxinline] with measurement operators [mathjaxinline]M_ k = \{ M_{k0}, M_{k1}\}[/mathjaxinline], where [mathjaxinline]M_{k0} = |\phi _ k\rangle \langle \phi _ k|[/mathjaxinline] and [mathjaxinline]M_{k1} = I - M_{k0}[/mathjaxinline], with </p>
<table id="a0000000020" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000021">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\phi _0\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \cos (\pi /8) |0\rangle + \sin (\pi /8) | 1\rangle[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.16)</td>
</tr>
<tr id="a0000000022">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |\phi _1\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \cos (\pi /8) |0\rangle - \sin (\pi /8) | 1\rangle \, .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.17)</td>
</tr>
</table>
<p>
Give the probability [mathjaxinline]p[/mathjaxinline] of Bob correctly obtaining [mathjaxinline]x_ k[/mathjaxinline]. Enter your result as a decimal number, correct to wihin 1%: </p>
<p>
<p style="display:inline">[mathjaxinline]p =[/mathjaxinline]</p>
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<p>
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</div></p>
</li>
<li value="2">
<p>
Is this possible using classical bits? It turns out no: for the best possible encoding of [mathjaxinline]m[/mathjaxinline] classical bits using [mathjaxinline]n[/mathjaxinline] classical bits, with readout error probability [mathjaxinline]1-p[/mathjaxinline] (assume [mathjaxinline]p\geq 1/2[/mathjaxinline]), it must be true that [mathjaxinline]n \ge f(m,H)[/mathjaxinline] for [mathjaxinline]f[/mathjaxinline] being a function of [mathjaxinline]m[/mathjaxinline] and the binary entropy function [mathjaxinline]H = H(p)[/mathjaxinline]. </p>
<p>
Give [mathjaxinline]f(m,H)[/mathjaxinline]. Remember to explicitly use [mathjaxinline]{\tt *}[/mathjaxinline] for multiplication. Enter your response using [mathjaxinline]{\tt H}[/mathjaxinline] and [mathjaxinline]{\tt m}[/mathjaxinline] as symbols, in addition to the usual standard functions and numbers. </p>
<p>
<p style="display:inline">[mathjaxinline]f(m,H) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_s12-wk10-raccesss_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_s12-wk10-raccesss_3_1" class="equation">`{::}`</div>
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</p>
<p>
Hint: let [mathjaxinline]X[/mathjaxinline] be the message sent, and [mathjaxinline]Y[/mathjaxinline] be the message received. Clearly, [mathjaxinline]n \geq I(X;Y) = H(X) - H(X|Y)[/mathjaxinline]. Find an expression for [mathjaxinline]H(X|Y)[/mathjaxinline], and use [mathjaxinline]H(X)\leq m[/mathjaxinline] to obtain the desired result. </p>
<p>
What is the bound when the numerical value of [mathjaxinline]p[/mathjaxinline] you found, above, is plugged in? (Note that your response should depend on [mathjaxinline]m[/mathjaxinline] as well.) </p>
<p>
<p style="display:inline">[mathjaxinline]n \ge[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_s12-wk10-raccesss_4_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_s12-wk10-raccesss_4_1" class="answer"/>
<div id="display_s12-wk10-raccesss_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_s12-wk10-raccesss_4_1_dynamath" name="input_s12-wk10-raccesss_4_1_dynamath"/>
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</p>
<p>
<div class="solution-span">
<span id="solution_s12-wk10-raccesss_solution_2"/>
</div></p>
</li>
<li value="3">
<p>
The random access code used in part (a) above can be improved to allow three bits to be encoded in one qubit. Show that this can be accomplished if Alice also uses the eigenstates of [mathjaxinline]Y[/mathjaxinline] in addition to those of [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline]. Give the probability [mathjaxinline]p[/mathjaxinline] of Bob obtaining the correct result, with this improved scheme. </p>
<p>
<p style="display:inline">[mathjaxinline]p =[/mathjaxinline]</p>
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<p>
<div class="solution-span">
<span id="solution_s12-wk10-raccesss_solution_3"/>
</div></p>
</li>
<li value="4">
<p>
(Optional) Construct a quantum key distribution scheme based on quantum random access codes, and give a good information-theoretic argument for its security. </p>
</li>
</ol>
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return;
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var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('div.course-index').length)){
return get_search_module_ficus();
}
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.371.3x+2T2018+type@vertical+block@Holevos_Theorem_I" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.371.3x+2T2018" data-request-token="6be83218886511ef879c0affec87671b" data-block-type="vertical" data-runtime-version="1" data-has-score="False" data-graded="True">
<h2 class="hd hd-2 unit-title">Holevo's Theorem I</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-holevo">
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<div id="problem_s12-wk10-holevo" class="problems-wrapper" role="group"
aria-labelledby="s12-wk10-holevo-problem-title"
data-problem-id="block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-holevo" data-url="/courses/course-v1:MITx+8.371.3x+2T2018/xblock/block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-holevo/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
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<h3 class="hd hd-3 problem-header" id="s12-wk10-holevo-problem-title" aria-describedby="block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-holevo-problem-progress" tabindex="-1">
Holevo&#39;s Theorem I
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-holevo-problem-progress"></div>
<div class="problem">
<div>
<p>
Suppose Alice sends Bob a stream of symbols (each a single qubit) chosen equally likely from the four pure states </p>
<table id="a0000000026" class="eqnarray" cellspacing="0" cellpadding="7" width="100%" style="table-layout:auto">
<tr id="a0000000027">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |X_1\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle |0\rangle[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.20)</td>
</tr>
<tr id="a0000000028">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |X_2\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt{\frac{1}{3}}\left[{ |0\rangle - \sqrt{2} |1\rangle }\right][/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.21)</td>
</tr>
<tr id="a0000000029">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |X_3\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt{\frac{1}{3}} \left[{ |0\rangle - \sqrt{2} e^{2\pi i/3} |1\rangle }\right][/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.22)</td>
</tr>
<tr id="a0000000030">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |X_4\rangle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt{\frac{1}{3}} \left[{ |0\rangle - \sqrt{2} e^{4\pi i/3} |1\rangle }\right]\, .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.23)</td>
</tr>
</table>
<ol class="enumerate">
<li value="1">
<p>
Let [mathjaxinline]X[/mathjaxinline] describe Alice's classical source distribution. What is [mathjaxinline]H(X)[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]H(X) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_s12-wk10-holevo_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_s12-wk10-holevo_2_1" id="input_s12-wk10-holevo_2_1" aria-describedby="status_s12-wk10-holevo_2_1" value="" size="40"/>
<span class="trailing_text" id="trailing_text_s12-wk10-holevo_2_1"/>
<span class="status unanswered" id="status_s12-wk10-holevo_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_s12-wk10-holevo_2_1" class="answer"/>
</div>
</div></div>
</p>
</li>
<li value="2">
<p>
What is the density matrix [mathjaxinline]\rho[/mathjaxinline] describing Alice's message source? </p>
<p>
Enter your response as a list of lists of numbers, eg [mathjaxinline]{\tt [[1,0],[3,4/5]]}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\rho =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_s12-wk10-holevo_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_s12-wk10-holevo_3_1" id="input_s12-wk10-holevo_3_1" aria-describedby="status_s12-wk10-holevo_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_s12-wk10-holevo_3_1"/>
<span class="status unanswered" id="status_s12-wk10-holevo_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_s12-wk10-holevo_3_1" class="answer"/>
<div id="display_s12-wk10-holevo_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_s12-wk10-holevo_3_1_dynamath" name="input_s12-wk10-holevo_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_s12-wk10-holevo_solution_1"/>
</div></p>
</li>
<li value="3">
<p>
From Holevo's theorem, the maximum mutual information [mathjaxinline]I(X;Y)[/mathjaxinline] between Bob's received message [mathjaxinline]Y[/mathjaxinline] and Alice's transmission [mathjaxinline]X[/mathjaxinline] is given by </p>
<table id="a0000000043" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]I(X;Y) \leq S(\rho ) - \sum _{x} p_{x} S(\rho _{x})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.34)</td>
</tr>
</table>
<p>
for [mathjaxinline]\rho _ x = |X_ x\rangle \langle X_ x|[/mathjaxinline]. </p>
<p>
Give a numerical value for this Holevo bound: </p>
<p>
<p style="display:inline">[mathjaxinline]I(X;Y) \leq[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_s12-wk10-holevo_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_s12-wk10-holevo_4_1" id="input_s12-wk10-holevo_4_1" aria-describedby="status_s12-wk10-holevo_4_1" value="" size="40"/>
<span class="trailing_text" id="trailing_text_s12-wk10-holevo_4_1"/>
<span class="status unanswered" id="status_s12-wk10-holevo_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_s12-wk10-holevo_4_1" class="answer"/>
</div>
</div></div>
</p>
</li>
</ol>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Holevo&#39;s Theorem 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_s12-wk10-holevo" >
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</button>
<div class="submission-feedback" id="submission_feedback_s12-wk10-holevo">
<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">
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<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
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is-hidden"
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<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
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</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.3x+2T2018+type@html+block@html_site_search_box2xxxx">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.3x+2T2018+type@html+block@html_site_search_box2xxxx" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.371.3x+2T2018" data-request-token="6be83218886511ef879c0affec87671b" data-block-type="html" data-runtime-version="1" data-has-score="False" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<span><a href="/asset-v1:MITx+8.371.3x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.3x+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+8.371.3x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.3x+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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.371.3x+2T2018+type@vertical+block@Holevos_Theorem_II" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.371.3x+2T2018" data-request-token="6be83218886511ef879c0affec87671b" data-block-type="vertical" data-runtime-version="1" data-has-score="False" data-graded="True">
<h2 class="hd hd-2 unit-title">Holevo's Theorem II</h2>
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<h3 class="hd hd-3 problem-header" id="s12-wk10-holevo2-problem-title" aria-describedby="block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-holevo2-problem-progress" tabindex="-1">
Holevo&#39;s Theorem II
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.3x+2T2018+type@problem+block@s12-wk10-holevo2-problem-progress"></div>
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<ol class="enumerate">
<li value="1">
<p>
Bob receives his message by implementing a quantum measurement, which turns Alice's transmitted qubits into one or more classical bits. </p>
<p>
One way he could do this to implement a generalized quantum measurement scheme, known as a positive operator-valued measurement (&#8220;POVM"). This is basically a specification of a general quantum operation in terms of products of the pairs of operation elements involved in the operator sum representation expression for the operation. </p>
<p>
There exists a POVM which achieves [mathjaxinline]I_0 = I(X;Y) \approx 0.415[/mathjaxinline] bits. </p>
<p>
This POVM is specified by a set of operators [mathjaxinline]M_ k[/mathjaxinline] such that [mathjaxinline]\sum _ k M_ k = I[/mathjaxinline]. For [mathjaxinline]\rho[/mathjaxinline] being the density matrix describing Alice's source, the measurement result [mathjaxinline]y = {\rm tr}(\rho M_ k)[/mathjaxinline] has mutual information [mathjaxinline]I_0[/mathjaxinline] with Alice's transmission [mathjaxinline]|X_ x\rangle[/mathjaxinline], of her classical message [mathjaxinline]x[/mathjaxinline]. </p>
<p>
Each [mathjaxinline]M_ k[/mathjaxinline] can be expressed as </p>
<table id="a0000000044" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]M_ k = \frac{1}{2} |\phi _ k\rangle \langle \phi _ k| \, ,[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.35)</td>
</tr>
</table>
<p>
and we may choose [mathjaxinline]|\phi _1\rangle = |1\rangle[/mathjaxinline]. Give the other three states which define [mathjaxinline]M_2[/mathjaxinline], [mathjaxinline]M_3[/mathjaxinline], and [mathjaxinline]M_4[/mathjaxinline], such that the POVM achieves the specified mutual information. </p>
<p>
<span>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">[mathjaxinline]|\phi _2\rangle =[/mathjaxinline] <div id="inputtype_s12-wk10-holevo2_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MathjaxPreprocessorForQM"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/dc8916af8ae257e4388dae38f570cb8b/asset-v1:MITx+8.371.3x+2T2018+type@asset+block/mathjax_preprocessor_for_QM_subtext.js"/>
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<div id="display_s12-wk10-holevo2_2_1" class="equation">`{::}`</div>
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<br/>
<p style="display:inline">[mathjaxinline]|\phi _3\rangle =[/mathjaxinline] <div id="inputtype_s12-wk10-holevo2_2_2" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MathjaxPreprocessorForQM"/>
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<div id="display_s12-wk10-holevo2_2_2" class="equation">`{::}`</div>
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<br/>
<p style="display:inline">[mathjaxinline]|\phi _4\rangle =[/mathjaxinline] <div id="inputtype_s12-wk10-holevo2_2_3" class="text-input-dynamath capa_inputtype inline textline">
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<p id="answer_s12-wk10-holevo2_2_3" class="answer"/>
<div id="display_s12-wk10-holevo2_2_3" class="equation">`{::}`</div>
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<span id="solution_s12-wk10-holevo2_solution_1"/>
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</li>
<li value="2">
<p>
(optional) Give a short and understandble proof that the Holevo bound is not achievable in this scenario. </p>
</li>
</ol>
<p>
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<span id="solution_s12-wk10-holevo2_solution_2"/>
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