{"id":4142,"date":"2024-09-22T15:05:48","date_gmt":"2024-09-22T15:05:48","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4142"},"modified":"2024-09-24T11:29:58","modified_gmt":"2024-09-24T11:29:58","slug":"entropy-of-composite-systems","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/entropy-of-composite-systems\/","title":{"rendered":"Entropy of composite systems"},"content":{"rendered":"\n

Some of the properties of the von Neumann entropy for composite systems<\/p>\n\n\n\n

are similar to those of Shannon entropy, while some others are quite di\ufb00erent.<\/p>\n\n\n\n

We discuss a few here.<\/p>\n\n\n\n

1. Concavity: S(\u03c1) is a concave function. That is, for a linear combination<\/p>\n\n\n\n

of states \u03c1 = c<\/p>\n\n\n\n

1<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

+ c<\/p>\n\n\n\n

2<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

, the resulting entropy is usually greater than the<\/p>\n\n\n\n

weighted sum of the individual entropies:<\/p>\n\n\n\n

S(\u03c1) \u2265 c<\/p>\n\n\n\n

1<\/p>\n\n\n\n

S<\/p>\n\n\n\n

\ue000<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

\ue001<\/p>\n\n\n\n

+ c<\/p>\n\n\n\n

2<\/p>\n\n\n\n

S<\/p>\n\n\n\n

\ue000<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

\ue001<\/p>\n\n\n\n

. (11.35)<\/p>\n\n\n\n

The physical interpretation is that as when two systems are mixed, the re-<\/p>\n\n\n\n

sultant is more uniform than each of the individual systems. To prove this,<\/p>\n\n\n\n

you need to remember that the logarithm is not a linear function. It is, in<\/p>\n\n\n\n

fact, a concave function (look at the graph of Figure 11.2). This also means<\/p>\n\n\n\n

that the function x log x is concave. In the basis {|ii} in which \u03c1 is diagonal,<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

i<\/p>\n\n\n\n

= hi|\u03c1|ii. Let\u2019s introduce the notation \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

i<\/p>\n\n\n\n

= hi|\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

|ii etc.<\/p>\n\n\n\n

Proof.<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

i<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

i<\/p>\n\n\n\n

\u2265 c<\/p>\n\n\n\n

1<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

i<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

i<\/p>\n\n\n\n

+ c<\/p>\n\n\n\n

2<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

i<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

i<\/p>\n\n\n\n

=\u21d2 S(\u03c1) = \u2212<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

i<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

i<\/p>\n\n\n\n

\u2265 \u2212<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

(c<\/p>\n\n\n\n

1<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

i<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

i<\/p>\n\n\n\n

+ c<\/p>\n\n\n\n

2<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

i<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

i<\/p>\n\n\n\n

)<\/p>\n\n\n\n

= c<\/p>\n\n\n\n

1<\/p>\n\n\n\n

S(\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

) + c<\/p>\n\n\n\n

2<\/p>\n\n\n\n

S(\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

).<\/p>\n\n\n\n

2. Quantum relative entropy. Suppose that {|ii} and {|mi} are two sets<\/p>\n\n\n\n

of orthogonal bases for the Hilbert space of the system. For density operators<\/p>\n\n\n\n

\u03c1 =<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

|iihi|; \u03c3 =<\/p>\n\n\n\n

X<\/p>\n\n\n\n

m<\/p>\n\n\n\n

q<\/p>\n\n\n\n

m<\/p>\n\n\n\n

|mihm|,<\/p>\n\n\n\n

\"\"<\/p>\n\n\n\n

Characterization of Quantum Information 227<\/p>\n\n\n\n

we can de\ufb01ne the relative entropy as<\/p>\n\n\n\n

S(\u03c1 k \u03c3) = Tr<\/p>\n\n\n\n

\ue002<\/p>\n\n\n\n

\u03c1(log \u03c1 \u2212 log \u03c3)<\/p>\n\n\n\n

\ue003<\/p>\n\n\n\n

. (11.36)<\/p>\n\n\n\n

In evaluating this quantity, we \ufb01nd that it is always non-negative: a result<\/p>\n\n\n\n

sometimes known as Klein\u2019s inequality.<\/p>\n\n\n\n

Proof.<\/p>\n\n\n\n

S(\u03c1 k \u03c3) =<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

hi|\u03c1(log \u03c1 \u2212 log \u03c3)|ii<\/p>\n\n\n\n

=<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

log p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

\u2212 p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

hi|log \u03c3|ii. (11.37)<\/p>\n\n\n\n

Here, hi|log \u03c3|ii = hi|<\/p>\n\n\n\n

X<\/p>\n\n\n\n

m<\/p>\n\n\n\n

log q<\/p>\n\n\n\n

m<\/p>\n\n\n\n

|mihm|ii<\/p>\n\n\n\n

=<\/p>\n\n\n\n

X<\/p>\n\n\n\n

m<\/p>\n\n\n\n

log q<\/p>\n\n\n\n

m<\/p>\n\n\n\n

P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

(11.38)<\/p>\n\n\n\n

where P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

\u2261 hi|mihm|ii (11.39)<\/p>\n\n\n\n

\u2265 0;<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

= 1 =<\/p>\n\n\n\n

X<\/p>\n\n\n\n

m<\/p>\n\n\n\n

P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

(11.40)<\/p>\n\n\n\n

(Such a matrix is called doubly stochastic.)<\/p>\n\n\n\n

So, S(\u03c1 k \u03c3) =<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

“<\/p>\n\n\n\n

log p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

\u2212<\/p>\n\n\n\n

X<\/p>\n\n\n\n

m<\/p>\n\n\n\n

P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

log q<\/p>\n\n\n\n

m<\/p>\n\n\n\n

#<\/p>\n\n\n\n

(11.41)<\/p>\n\n\n\n

=<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i,m<\/p>\n\n\n\n

p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

log<\/p>\n\n\n\n

p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

q<\/p>\n\n\n\n

m<\/p>\n\n\n\n

(since<\/p>\n\n\n\n

X<\/p>\n\n\n\n

m<\/p>\n\n\n\n

P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

= 1)<\/p>\n\n\n\n

\u2265<\/p>\n\n\n\n

X<\/p>\n\n\n\n

i,m<\/p>\n\n\n\n

p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

P<\/p>\n\n\n\n

im<\/p>\n\n\n\n

\ue012<\/p>\n\n\n\n

1 \u2212<\/p>\n\n\n\n

q<\/p>\n\n\n\n

m<\/p>\n\n\n\n

p<\/p>\n\n\n\n

i<\/p>\n\n\n\n

\ue013<\/p>\n\n\n\n

(since log x \u2265 1 \u2212<\/p>\n\n\n\n

1<\/p>\n\n\n\n

x<\/p>\n\n\n\n

)<\/p>\n\n\n\n

= 0, (using Eq 11.41)<\/p>\n\n\n\n

=\u21d2 S(\u03c1 k \u03c3) \u2265 0. (11.42)<\/p>\n\n\n\n

3. Subadditivity. Given two systems A and B with joint state \u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

, and<\/p>\n\n\n\n

reduced density matrices \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

and \u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

, the joint entropy de\ufb01ned simply as<\/p>\n\n\n\n

S(\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

) \u2261 \u2212Tr\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

(11.43)<\/p>\n\n\n\n

satis\ufb01es<\/p>\n\n\n\n

S(\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

) \u2264 S(\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

) + S(\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

), (11.44)<\/p>\n\n\n\n

with equality only when the two systems are uncorrelated. Thus entanglement<\/p>\n\n\n\n

reduces the entropy, i.e., increases the information, of the system.<\/p>\n\n\n\n

<\/p>\n\n\n\n

228 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n

Proof. The proof follows as an application of Klein\u2019s inequality for \u03c1 = \u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

and \u03c3 = \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

\u2297 \u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

. Suppose |ii and |mi are bases for the Hilbert spaces of A<\/p>\n\n\n\n

and B, respectively. From Klein\u2019s inequality,<\/p>\n\n\n\n

S(\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

) \u2264 \u2212Tr\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

log(\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

\u2297 \u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

)<\/p>\n\n\n\n

= \u2212Tr\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

\u2212 Tr\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

Now the \ufb01rst term in this is<\/p>\n\n\n\n

\u2212hi, m|\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

|i, mi = \u2212Tr<\/p>\n\n\n\n

A<\/p>\n\n\n\n

\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

log \u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

= S(\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

).<\/p>\n\n\n\n

Similarly for the other term. So we have<\/p>\n\n\n\n

S(\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

) \u2264 S(\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

) + S(\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

) (11.45)<\/p>\n\n\n\n

There is another result, the triangle inequality also known as the Araki\u2013<\/p>\n\n\n\n

Lieb inequality, that can be similarly proved:<\/p>\n\n\n\n

S(\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

) \u2265 |S(\u03c1<\/p>\n\n\n\n

A<\/p>\n\n\n\n

) \u2212 S(\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

)|. (11.46)<\/p>\n\n\n\n

4. Conditional entropy.<\/p>\n\n\n\n

S(A|B) \u2261 S(\u03c1<\/p>\n\n\n\n

AB<\/p>\n\n\n\n

) \u2212 S(\u03c1<\/p>\n\n\n\n

B<\/p>\n\n\n\n

). (11.47)<\/p>\n\n\n\n

While Shannon conditional entropy can never be negative, the von Neumann<\/p>\n\n\n\n

entropy can, for systems that are entangled [16]. This can be proved to be a<\/p>\n\n\n\n

criterion for entanglement.<\/p>\n\n\n\n

There are many more inequalities and properties of the von Neumann<\/p>\n\n\n\n

entropy that can be proved, for which we refer you to Nielsen and Chuang<\/p>\n\n\n\n

[50], the book by Ohya and Petz [51] and the review article by Wehrl [71].<\/p>\n","protected":false},"excerpt":{"rendered":"

Some of the properties of the von Neumann entropy for composite systems are similar to those of Shannon entropy, while some others are quite di\ufb00erent. We discuss a few here. 1. Concavity: S(\u03c1) is a concave function. That is, for a linear combination of states \u03c1 = c 1 \u03c1 A + c 2 \u03c1 […]<\/p>\n","protected":false},"author":1,"featured_media":4045,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[500],"tags":[],"class_list":["post-4142","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-characterization-of-quantum-information"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-computing-2.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4142","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4142"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4142\/revisions"}],"predecessor-version":[{"id":4578,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4142\/revisions\/4578"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4045"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4142"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4142"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4142"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}