A given density operator could represent a pure or a mixed state. If the<\/p>\n\n\n\n
system is pure, then the state is a ray in Hilbert space, and the density operator<\/p>\n\n\n\n
can be expressed as<\/p>\n\n\n\n
\u03c1 = |\u03c8ih\u03c8|, for some |\u03c8i.<\/p>\n\n\n\n
Such a density matrix satis\ufb01es<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= |\u03c8ih\u03c8|\u03c8ih\u03c8| = \u03c1, (5.17)<\/p>\n\n\n\n
Tr(\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) = Tr(\u03c1) = 1. (5.18)<\/p>\n\n\n\n
This is not true if \u03c1 represents a mixed state, where<\/p>\n\n\n\n
\u03c1 =<\/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
for which \u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n,m<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
m<\/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
|\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
| 6= \u03c1. (5.19)<\/p>\n\n\n\n
<\/p>\n\n\n\n
84 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
In the orthonormal basis {|ii}, we have<\/p>\n\n\n\n
Tr(\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n,m<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
m<\/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
|\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
|ii<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,n,m<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
m<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
|iihi|\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
|\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n, mp<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
m<\/p>\n\n\n\n
|h\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
i|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2264<\/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
!<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1 (5.20)<\/p>\n\n\n\n
The equality holds only when h\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
m<\/p>\n\n\n\n
i = a pure phase for all pairs n and m,<\/p>\n\n\n\n
which means that the density matrix comprises only one state vector in Hilbert<\/p>\n\n\n\n
space: a pure state. In fact, the quantity Tr(\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) is sometimes called the purity<\/p>\n\n\n\n
of the state. A completely pure state has Tr(\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) = 1 and a completely mixed<\/p>\n\n\n\n
state has Tr(\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
n<\/p>\n\n\n\n
. These ideas will be very useful when we study quantum<\/p>\n\n\n\n
information theory. There we will also encounter the notion of entropy as a<\/p>\n\n\n\n
measure of information, which can also be used to distinguish pure and mixed<\/p>\n\n\n\n
states.<\/p>\n\n\n\n
Example 5.1.5. For the state pure state |+i,<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
p<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
!<\/p>\n\n\n\n
2<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
2 2<\/p>\n\n\n\n
2 2<\/p>\n\n\n\n
!<\/p>\n\n\n\n
= \u03c1,<\/p>\n\n\n\n
Tr\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
p<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1.<\/p>\n\n\n\n
For the unpolarized electron beam (Example 5.1.1), which is a maximally<\/p>\n\n\n\n
mixed state, we have<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
m<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
,<\/p>\n\n\n\n
Tr\u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
m<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n","protected":false},"excerpt":{"rendered":"
A given density operator could represent a pure or a mixed state. If the system is pure, then the state is a ray in Hilbert space, and the density operator can be expressed as \u03c1 = |\u03c8ih\u03c8|, for some |\u03c8i. Such a density matrix satis\ufb01es \u03c1 2 = |\u03c8ih\u03c8|\u03c8ih\u03c8| = \u03c1, (5.17) Tr(\u03c1 2 ) […]<\/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-4027","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\/4027","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=4027"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4027\/revisions"}],"predecessor-version":[{"id":4568,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4027\/revisions\/4568"}],"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=4027"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4027"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}