{"id":4140,"date":"2024-09-22T15:03:38","date_gmt":"2024-09-22T15:03:38","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4140"},"modified":"2024-09-22T15:03:39","modified_gmt":"2024-09-22T15:03:39","slug":"properties-of-the-von-neumann-entropy","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/properties-of-the-von-neumann-entropy\/","title":{"rendered":"Properties of the von Neumann entropy"},"content":{"rendered":"\n

Some properties of the von Neumann entropy immediately follow from the<\/p>\n\n\n\n

de\ufb01nition.<\/p>\n\n\n\n

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

Characterization of Quantum Information 225<\/p>\n\n\n\n

1. The minimum value of S(\u03c1), zero, occurs for pure states.<\/p>\n\n\n\n

S(\u03c1) \u2265 0. (11.29)<\/p>\n\n\n\n

Thus even though a pure state embodies probabilities of measurement out-<\/p>\n\n\n\n

comes, the information carried by it is zero since it represents a de\ufb01nite vector<\/p>\n\n\n\n

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

2. The maximum value of S(\u03c1) is log d, where d is the dimensionality of<\/p>\n\n\n\n

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

S(\u03c1) \u2264 log d. (11.30)<\/p>\n\n\n\n

This occurs for maximally mixed states with each \u03c1<\/p>\n\n\n\n

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

taking the value 1\/d.<\/p>\n\n\n\n

You will prove this in an exercise.<\/p>\n\n\n\n

3. Invariance under unitary transformations:<\/p>\n\n\n\n

Under unitary evolution U of the quantum system, the von Neumann entropy<\/p>\n\n\n\n

remains unchanged.<\/p>\n\n\n\n

S(U\u03c1U<\/p>\n\n\n\n

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

) = S(\u03c1). (11.31)<\/p>\n\n\n\n

4. Entropy of preparation:<\/p>\n\n\n\n

We can think of entropy as a measure of mixedness of the system, or its<\/p>\n\n\n\n

departure from purity. When constructing a state \u03c1 out of an ensemble of<\/p>\n\n\n\n

pure states |xi with probability p(x), in general we will \ufb01nd that<\/p>\n\n\n\n

H(X) \u2265 S(\u03c1). (11.32)<\/p>\n\n\n\n

That is, the Shannon (classical) entropy is greater than the von Neumann<\/p>\n\n\n\n

entropy. The equality (Equation 11.27) holds when the |xi are mutually or-<\/p>\n\n\n\n

thogonal. The interpretation of this result is that when viewed in a basis in<\/p>\n\n\n\n

which \u03c1 is not diagonal, we are not in the same basis in which the system was<\/p>\n\n\n\n

prepared. Measurement results in such a basis will have probabilities such that<\/p>\n\n\n\n

the entropy is more than the von Neumann entropy. The latter is therefore<\/p>\n\n\n\n

called the entropy of preparation of the system.<\/p>\n\n\n\n

Example 11.2.2. For a state that is 25% |0i and 75% |+i, the Shannon<\/p>\n\n\n\n

entropy is<\/p>\n\n\n\n

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

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

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

log 4 +<\/p>\n\n\n\n

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

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

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

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

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

= 0.81 bits.<\/p>\n\n\n\n

The density matrix is<\/p>\n\n\n\n

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

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

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

1 0<\/p>\n\n\n\n

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

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

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

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

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

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

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

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

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

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

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

5 3<\/p>\n\n\n\n

3 3<\/p>\n\n\n\n

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

with eigenvalues<\/p>\n\n\n\n

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

\/2 \u00b1<\/p>\n\n\n\n

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

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

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

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

\/2, so that the von Neumann entropy is<\/p>\n\n\n\n

S(\u03c1) = 0.485 qubits<\/p>\n","protected":false},"excerpt":{"rendered":"

Some properties of the von Neumann entropy immediately follow from the de\ufb01nition. Characterization of Quantum Information 225 1. The minimum value of S(\u03c1), zero, occurs for pure states. S(\u03c1) \u2265 0. (11.29) Thus even though a pure state embodies probabilities of measurement out- comes, the information carried by it is zero since it represents a […]<\/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-4140","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\/4140","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=4140"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4140\/revisions"}],"predecessor-version":[{"id":4141,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4140\/revisions\/4141"}],"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=4140"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4140"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4140"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}