{"id":4146,"date":"2024-09-22T15:09:12","date_gmt":"2024-09-22T15:09:12","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4146"},"modified":"2024-09-22T15:09:13","modified_gmt":"2024-09-22T15:09:13","slug":"fidelity","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/fidelity\/","title":{"rendered":"Fidelity"},"content":{"rendered":"\n

Another important measure for comparing probability distributions is the<\/p>\n\n\n\n

\ufb01delity, which is easily extended to quantum states. This is variously de\ufb01ned<\/p>\n\n\n\n

in di\ufb00erent texts, but we will stick to a simple operational de\ufb01nition here:<\/p>\n\n\n\n

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

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

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

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

p(x)q(x). (11.50)<\/p>\n\n\n\n

The square root is used so that we have F(p(x), p(x)) = 1. This de\ufb01nition<\/p>\n\n\n\n

is compatible with the inner product of two vectors with components {p(x)}<\/p>\n\n\n\n

and {q(x)}<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Characterization of Quantum Information 231<\/p>\n\n\n\n

In the quantum case, the \ufb01delity between a pure state |\u03c8i and a state |\u03c6i<\/p>\n\n\n\n

is the inner product:<\/p>\n\n\n\n

F(\u03c8, \u03c6) = h\u03c6|\u03c8i (11.51)<\/p>\n\n\n\n

|F|<\/p>\n\n\n\n

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

can also be thought of as the probability of confusing the state |\u03c8i with<\/p>\n\n\n\n

|\u03c6i in an experimental situation. Another way of looking at it is that if the<\/p>\n\n\n\n

state |\u03c8i is sent through a communication protocol, the probability that the<\/p>\n\n\n\n

end state |\u03c6i is the same as the input state is (the mod-square of) the \ufb01delity<\/p>\n\n\n\n

of the process. The \ufb01delity is minimum, 0, if the two states are orthogonal, and<\/p>\n\n\n\n

maximum, 1, if the two states are identical. Classically, these are the only two<\/p>\n\n\n\n

situations that could possibly arise. But in the quantum world, there exists<\/p>\n\n\n\n

a continuity of states connecting the two possibilities, and this distinguishes<\/p>\n\n\n\n

quantum information from classical.<\/p>\n\n\n\n

One can extend this de\ufb01nition to mixed states as well: for states \u03c1 and \u03c3,<\/p>\n\n\n\n

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

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

\u03c1\u03c3). (11.52)<\/p>\n\n\n\n

Example 11.3.2. If we have a pure state |\u03c8i and a mixed state \u03c1, we can<\/p>\n\n\n\n

calculate the \ufb01delity as<\/p>\n\n\n\n

F(|\u03c8i, \u03c1) = Tr(<\/p>\n\n\n\n

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

|\u03c8ih\u03c8|\u03c1)<\/p>\n\n\n\n

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

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

h\u03c8|\u03c1|\u03c8i)<\/p>\n\n\n\n

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

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

h\u03c8|\u03c1|\u03c8i (11.53)<\/p>\n\n\n\n

Example 11.3.3. If two density matrices \u03c1 and \u03c3 commute then they can be<\/p>\n\n\n\n

diagonalized in the same basis and the \ufb01delity can be calculated as<\/p>\n\n\n\n

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

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

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

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

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

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

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

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

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

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

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

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

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

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

p(x)q(x)|xihx| = F(p(x), q(x)). (11.54)<\/p>\n\n\n\n

Fidelity is not a distance, but can be used to de\ufb01ne one between density<\/p>\n\n\n\n

operators, the so-called Bures distance<\/p>\n\n\n\n

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

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

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

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

2 \u2212 2F, (11.55)<\/p>\n\n\n\n

which is a metric on the space of states.<\/p>\n","protected":false},"excerpt":{"rendered":"

Another important measure for comparing probability distributions is the \ufb01delity, which is easily extended to quantum states. This is variously de\ufb01ned in di\ufb00erent texts, but we will stick to a simple operational de\ufb01nition here: F(p(x), q(x)) = X x p p(x)q(x). (11.50) The square root is used so that we have F(p(x), p(x)) = 1. […]<\/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-4146","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\/4146","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=4146"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4146\/revisions"}],"predecessor-version":[{"id":4147,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4146\/revisions\/4147"}],"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=4146"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4146"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}