The density operator on a Hilbert space, de\ufb01ned by Equation 5.4 satis\ufb01es<\/p>\n\n\n\n
the following properties:<\/p>\n\n\n\n
1. \u02c6\u03c1 is Hermitian.<\/p>\n\n\n\n
Proof: \u02c6\u03c1<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
| = \u02c6\u03c1. (5.11)<\/p>\n\n\n\n
2. \u02c6\u03c1 is non-negative, that is, for any vector |vi, hv|\u02c6\u03c1|vi \u2265 0. (This translates<\/p>\n\n\n\n
to its eigenvalues being non-negative, or det(\u03c1) \u2265 0.)<\/p>\n\n\n\n
Proof: hv|\u02c6\u03c1|vi =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
hv|p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|vi<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|hv|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2265 0 (5.12)<\/p>\n\n\n\n
since the right side is a sum of numbers that are always positive or zero.<\/p>\n\n\n\n
3. It satis\ufb01es Tr\u02c6\u03c1 = 1.<\/p>\n\n\n\n
Proof: In an orthonormal basis {|ii},<\/p>\n\n\n\n
Tr\u03c1 =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
hi|<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|<\/p>\n\n\n\n
!<\/p>\n\n\n\n
|ii<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
hi|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|ii<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|iihi|<\/p>\n\n\n\n
!<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i = 1 (5.13)<\/p>\n\n\n\n
In general, any operator on a Hilbert space satisfying these properties is<\/p>\n\n\n\n
de\ufb01ned as a density operator and can be used to predict the probabilities of<\/p>\n\n\n\n
outcomes of measurement on the system, bypassing the state-vector formalism<\/p>\n\n\n\n
altogether.<\/p>\n\n\n\n
Example 5.1.4. For a system described by continuous variables, for example<\/p>\n\n\n\n
position x, the density operator will be expressed as<\/p>\n\n\n\n
\u03c1 =<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
dx dx<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u03c9(x, x<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)|xihx<\/p>\n\n\n\n
Mixed States, Open Systems, and the Density Operator 83<\/p>\n\n\n\n
For a pure state, we will have<\/p>\n\n\n\n
\u03c1 =<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
dx dx<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u03c8(x)\u03c8<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
(x<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)|xihx<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|, (5.15)<\/p>\n\n\n\n
where \u03c8(x) = hx|\u03c8i.<\/p>\n\n\n\n
Another property of the density operator that will be useful to us is con-<\/p>\n\n\n\n
vexity:<\/p>\n\n\n\n
De\ufb01nition 5.3. Convexity: A set of operators {\u02c6\u03c1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
} form a convex set if<\/p>\n\n\n\n
\u03c1 = \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
, 0 < \u03bb < 1, (5.16)<\/p>\n\n\n\n
for every pair \u03c1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
, \u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2208 {\u03c1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
}.<\/p>\n\n\n\n
Convexity has a very simple meaning: any two members of a convex set<\/p>\n\n\n\n
can be connected by a straight line without leaving the set. (See Figure 5.1.)<\/p>\n\n\n\n
FIGURE 5.1: (a) A convex set, (b) A non-convex set.<\/p>\n","protected":false},"excerpt":{"rendered":"
The density operator on a Hilbert space, de\ufb01ned by Equation 5.4 satis\ufb01es the following properties: 1. \u02c6\u03c1 is Hermitian. Proof: \u02c6\u03c1 \u2020 = X n p \u2217 n |\u03c8 n i \u2020 h\u03c8 n | \u2020 = X n p n |\u03c8 n ih\u03c8 n | = \u02c6\u03c1. (5.11) 2. \u02c6\u03c1 is non-negative, that is, […]<\/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-4025","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\/4025","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=4025"}],"version-history":[{"count":3,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4025\/revisions"}],"predecessor-version":[{"id":4567,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4025\/revisions\/4567"}],"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=4025"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4025"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4025"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}