{"id":4144,"date":"2024-09-22T15:07:01","date_gmt":"2024-09-22T15:07:01","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4144"},"modified":"2024-09-22T15:07:02","modified_gmt":"2024-09-22T15:07:02","slug":"distance-measures","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/distance-measures\/","title":{"rendered":"Distance Measures"},"content":{"rendered":"\n

An important consideration in information theory is the comparison of two<\/p>\n\n\n\n

systems: probability distributions in the classical context and states (pure or<\/p>\n\n\n\n

mixed) in the quantum. For such comparisons, various measures collectively<\/p>\n\n\n\n

labeled distance measures have been proposed. We\u2019ll consider some of them<\/p>\n\n\n\n

here, to educate ourselves in the concepts involved.<\/p>\n\n\n\n

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

Characterization of Quantum Information 229<\/p>\n\n\n\n

11.3.1 Kolmogorov or trace distance<\/p>\n\n\n\n

A sort of distance between two probability distributions p(x) and q(x) for<\/p>\n\n\n\n

the same random variable X can be de\ufb01ned as<\/p>\n\n\n\n

D(p(x), q(x)) =<\/p>\n\n\n\n

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

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

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

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

|p(x) \u2212 q(x)|. (11.48)<\/p>\n\n\n\n

This is similar to a \u201cmetric\u201d for determining the distance between points in a<\/p>\n\n\n\n

space.<\/p>\n\n\n\n

One context in which such a measure is useful is in a dynamic process,<\/p>\n\n\n\n

where information X is sent through a (noisy) channel and appears as Y .<\/p>\n\n\n\n

We wish to compute the probability of error in the channel by comparing the<\/p>\n\n\n\n

two distributions. To do this, we \ufb01rst make a copy of the input and call it<\/p>\n\n\n\n

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

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

, and then look at the probability distribution of the pairs (X<\/p>\n\n\n\n

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

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

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

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

, Y ). Let\u2019s compute the trace distance between these two distributions<\/p>\n\n\n\n

p(x) = p(X<\/p>\n\n\n\n

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

= x, X = x) and q<\/p>\n\n\n\n

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

= p(X<\/p>\n\n\n\n

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

= x, Y = y):<\/p>\n\n\n\n

D(p, q) =<\/p>\n\n\n\n

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

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

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

x,y<\/p>\n\n\n\n

|p(x) \u2212 q(y)|<\/p>\n\n\n\n

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

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

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

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

x6=y<\/p>\n\n\n\n

p(x) +<\/p>\n\n\n\n

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

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

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

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

|p(x) \u2212 q(x)|<\/p>\n\n\n\n

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

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

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

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

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

6= Y ) + 1 \u2212 p(X<\/p>\n\n\n\n

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

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

= p(X 6= Y )<\/p>\n\n\n\n

For two quantum states \u03c1 and \u03c3, we can de\ufb01ne the Kolmogorov distance<\/p>\n\n\n\n

using the trace function<\/p>\n\n\n\n

D(\u03c1, \u03c3) =<\/p>\n\n\n\n

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

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

Tr|\u03c1 \u2212 \u03c3|. (11.49)<\/p>\n\n\n\n

How do we compute this? We will de\ufb01ne the mod of a matrix A by<\/p>\n\n\n\n

|A| =<\/p>\n\n\n\n

\u221a<\/p>\n\n\n\n

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

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

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

\u221a<\/p>\n\n\n\n

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

\u2020<\/p>\n\n\n\n

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

where the last equality holds if A is Hermitian, which is true of density ma-<\/p>\n\n\n\n

trices. We can easily see how this reduces to the classical distance, if we can<\/p>\n\n\n\n

diagonalize \u03c1 and \u03c3 in the same basis to write<\/p>\n\n\n\n

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

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

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

p(x)|xihx|, \u03c3 =<\/p>\n\n\n\n

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

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

q(x)|xihx|.<\/p>\n\n\n\n

Two matrices can be simultaneously diagonalized if and only if they commute.<\/p>\n\n\n\n

Then we see that<\/p>\n\n\n\n

D(\u03c1, \u03c3) =<\/p>\n\n\n\n

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

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

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

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

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

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

(p(x) \u2212 q(x))|xihx|<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

\ue00c<\/p>\n\n\n\n

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

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

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

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

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

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

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

|p(x) \u2212 q(x)| Tr|(|xihx|)|<\/p>\n\n\n\n

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

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

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

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

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

|p(x) \u2212 q(x)|<\/p>\n\n\n\n

= D(p(x), q(x)).<\/p>\n\n\n\n

Example 11.3.1. It may be instructive to visualize the trace distance between<\/p>\n\n\n\n

single qubits by a Bloch sphere picture. Let our states be represented by Bloch<\/p>\n\n\n\n

vectors ~p and ~q:<\/p>\n\n\n\n

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

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

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

( + ~p \u00b7 ~\u03c3) , \u03c3 =<\/p>\n\n\n\n

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

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

( + ~q \u00b7 ~\u03c3) .<\/p>\n\n\n\n

The trace distance is then<\/p>\n\n\n\n

D(\u03c1, \u03c3) =<\/p>\n\n\n\n

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

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

Tr|(~p \u2212 ~q) \u00b7 ~\u03c3|<\/p>\n\n\n\n

The matrix (~p \u2212 ~q) \u00b7 ~\u03c3 = ~a.~\u03c3 has eigenvalues \u00b1a. So the eigenvalues of |~a.~\u03c3|<\/p>\n\n\n\n

are |a| and Tr|~\u03c3| = 2|a|. So we have<\/p>\n\n\n\n

D(\u03c1, \u03c3) =<\/p>\n\n\n\n

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

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

(|~p \u2212 ~q|)<\/p>\n\n\n\n

which is half of the geometric distance between the points ~p and ~q in the<\/p>\n\n\n\n

Bloch ball.<\/p>\n\n\n\n

The trace distance can be interpreted as follows: if two quantum states<\/p>\n\n\n\n

are close in trace distance, then when measurements are performed in those<\/p>\n\n\n\n

states, the resulting probability distributions are close in the classical trace<\/p>\n\n\n\n

distance.<\/p>\n","protected":false},"excerpt":{"rendered":"

An important consideration in information theory is the comparison of two systems: probability distributions in the classical context and states (pure or mixed) in the quantum. For such comparisons, various measures collectively labeled distance measures have been proposed. We\u2019ll consider some of them here, to educate ourselves in the concepts involved. Characterization of Quantum Information […]<\/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-4144","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\/4144","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=4144"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4144\/revisions"}],"predecessor-version":[{"id":4145,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4144\/revisions\/4145"}],"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=4144"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4144"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}