{"id":4031,"date":"2024-09-19T21:55:53","date_gmt":"2024-09-19T21:55:53","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4031"},"modified":"2024-09-19T21:55:54","modified_gmt":"2024-09-19T21:55:54","slug":"decomposition-of-the-density-operator","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/19\/decomposition-of-the-density-operator\/","title":{"rendered":"Decomposition of the density operator"},"content":{"rendered":"\n

Often the density operator is the primary descriptor of a state. The de-<\/p>\n\n\n\n

composition in terms of component states<\/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|,<\/p>\n\n\n\n

is not always unique.<\/p>\n\n\n\n

For a pure state, it must be obvious that there is only one such decompo-<\/p>\n\n\n\n

sition, and this can be proved from the de\ufb01nitions:<\/p>\n\n\n\n

Theorem 5.1. For a pure state, there is a unique decomposition of \u02c6\u03c1 in the<\/p>\n\n\n\n

form of Equation 5.4, and in fact that decomposition consists of only one term<\/p>\n\n\n\n

Proof. We can see this by invoking the convexity property (Equation 5.16).<\/p>\n\n\n\n

Suppose our pure state density matrix admits such a decomposition<\/p>\n\n\n\n

\u02c6\u03c1<\/p>\n\n\n\n

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

= \u03bb|\u03c8<\/p>\n\n\n\n

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

ih\u03c8<\/p>\n\n\n\n

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

| + (1 \u2212 \u03bb)|\u03c8<\/p>\n\n\n\n

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

ih\u03c8<\/p>\n\n\n\n

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

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

= \u03bb\u03c1<\/p>\n\n\n\n

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

+ (1 \u2212 \u03bb)\u03c1<\/p>\n\n\n\n

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

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

Now since the state is pure, there exists some vector |ui such that<\/p>\n\n\n\n

\u02c6\u03c1<\/p>\n\n\n\n

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

= |uihu|.<\/p>\n\n\n\n

Consider an orthogonal vector |vi : hu|vi = 0.<\/p>\n\n\n\n

=\u21d2 hv|\u02c6\u03c1<\/p>\n\n\n\n

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

|vi = hv|uihu|vi = 0.<\/p>\n\n\n\n

=\u21d2 \u03bbhv| \u02c6\u03c1<\/p>\n\n\n\n

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

|vi + (1 \u2212 \u03bb)hv| \u02c6\u03c1<\/p>\n\n\n\n

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

|vi = 0.<\/p>\n\n\n\n

Since \u03bb and (1\u2212\u03bb) are positive, this equation can only be satis\ufb01ed if hv| \u02c6\u03c1<\/p>\n\n\n\n

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

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

0 = hv| \u02c6\u03c1<\/p>\n\n\n\n

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

|vi. This means \u02c6\u03c1<\/p>\n\n\n\n

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

and \u02c6\u03c1<\/p>\n\n\n\n

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

are orthogonal to |vi. But |vi can be any<\/p>\n\n\n\n

vector orthogonal to |ui. So we must have<\/p>\n\n\n\n

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

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

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

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

= \u02c6\u03c1.<\/p>\n\n\n\n

On the other hand, a mixed state \u03c1 has no unique decomposition in terms<\/p>\n\n\n\n

of pure states! This is easiest to see in our example of an unpolarized beam<\/p>\n\n\n\n

with density matrix<\/p>\n\n\n\n

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

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

: on subjecting this beam to SG tests along z or x or<\/p>\n\n\n\n

y or any other direction \u02c6n, it yields equal proportions of |\u2191i and |\u2193i states.<\/p>\n\n\n\n

This means that it can equally well be represented as equal parts of |0i and<\/p>\n\n\n\n

|1i, or |\u2191<\/p>\n\n\n\n

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

i and |\u2193<\/p>\n\n\n\n

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

i or even |\u2191<\/p>\n\n\n\n

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

i and |\u2193<\/p>\n\n\n\n

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

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

Exercise 5.8. Show that the density matrix<\/p>\n\n\n\n

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

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

can be expressed as<\/p>\n\n\n\n

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

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

(|\u2191ih\u2191| +<\/p>\n\n\n\n

|\u2193ih\u2193|) in any basis.<\/p>\n\n\n\n

In fact we can see from the convexity property of density matrices (5.16)<\/p>\n\n\n\n

that a given density operator \u03c1 can be expressed in in\ufb01nitely many ways in<\/p>\n\n\n\n

that form, so that it is impossible to identify any unique component density<\/p>\n\n\n\n

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

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

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

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

. For example, in the case of a single qubit state, three<\/p>\n\n\n\n

di\ufb00erent decompositions in terms of pure states that sit on the surface of the<\/p>\n\n\n\n

Bloch sphere, are shown in Figure 5.3. An in\ufb01nite number of such decompo-<\/p>\n\n\n\n

sitions is possible by choosing di\ufb00erent chords.<\/p>\n\n\n\n

Suppose that we prepare a mixed state \u03c1 with pure states |\u03c8<\/p>\n\n\n\n

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

i in certain<\/p>\n\n\n\n

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

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

, of the form in Equation 5.4. The p<\/p>\n\n\n\n

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

\u2019s in the density matrix<\/p>\n\n\n\n

represent the probability of \ufb01nding the state in |\u03c8<\/p>\n\n\n\n

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

i. However, when this state<\/p>\n\n\n\n

is passed on to someone who doesn\u2019t know how it was prepared, there is no<\/p>\n\n\n\n

way they can tell which states were used to prepare the system. Therefore,<\/p>\n\n\n\n

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

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

\u2019s can no longer be interpreted as probability of being in state |\u03c8<\/p>\n\n\n\n

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

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

since the decomposition is not unique. For this reason, it is not possible to<\/p>\n\n\n\n

interpret the eigenvalues of a density matrix as physical probabilities of the<\/p>\n\n\n\n

system being in particular states.<\/p>\n\n\n\n

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

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

FIGURE 5.3: Density matrix \u03c1 allowing three di\ufb00erent decompositions<\/p>\n","protected":false},"excerpt":{"rendered":"

Often the density operator is the primary descriptor of a state. The de- composition in terms of component states \u03c1 = X i p i |iihi|, is not always unique. For a pure state, it must be obvious that there is only one such decompo- sition, and this can be proved from the de\ufb01nitions: Theorem […]<\/p>\n","protected":false},"author":1,"featured_media":4020,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[491],"tags":[],"class_list":["post-4031","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-5-grand-unification"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4031","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=4031"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4031\/revisions"}],"predecessor-version":[{"id":4032,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4031\/revisions\/4032"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4020"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}