{"id":4029,"date":"2024-09-19T21:54:31","date_gmt":"2024-09-19T21:54:31","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4029"},"modified":"2024-09-19T21:54:32","modified_gmt":"2024-09-19T21:54:32","slug":"the-bloch-ball-and-the-density-operator","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/19\/the-bloch-ball-and-the-density-operator\/","title":{"rendered":"The Bloch ball and the density operator"},"content":{"rendered":"\n

The representation of a single qubit state on the Bloch sphere can be ex-<\/p>\n\n\n\n

tended to the density operator. The Bloch sphere is parametrized by spherical<\/p>\n\n\n\n

angles or in terms of the Bloch vector of Equation 4.2, which characterizes<\/p>\n\n\n\n

the polarization of the state. Can we use this kind of description for a mixed<\/p>\n\n\n\n

state?<\/p>\n\n\n\n

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

Mixed States, Open Systems, and the Density Operator 85<\/p>\n\n\n\n

For a single qubit state, \u03c1 is a 2 \u00d7 2 matrix and we can represent it as a<\/p>\n\n\n\n

linear combination of the Pauli spin matrices and the identity:<\/p>\n\n\n\n

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

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

+ ~a \u00b7 ~\u03c3, ~a \u2261 (a<\/p>\n\n\n\n

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

, a<\/p>\n\n\n\n

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

, a<\/p>\n\n\n\n

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

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

Since \u03c1 is Hermitian, we need a<\/p>\n\n\n\n

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

and a<\/p>\n\n\n\n

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

to be real. Since Tr(\u03c1) = 1 and the<\/p>\n\n\n\n

Pauli matrices are traceless, we must have p<\/p>\n\n\n\n

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

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

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

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

. Thus, if p<\/p>\n\n\n\n

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

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

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

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

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

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

, we can<\/p>\n\n\n\n

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

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

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

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

( + ~p \u00b7 ~\u03c3) =<\/p>\n\n\n\n

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

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

“<\/p>\n\n\n\n

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

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

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

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

\u2212 ip<\/p>\n\n\n\n

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

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

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

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

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

1 \u2212 p<\/p>\n\n\n\n

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

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

. (5.21)<\/p>\n\n\n\n

Since \u03c1 must be positive, we need det \u03c1 \u2265 0.<\/p>\n\n\n\n

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

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

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

(1 \u2212 ~p<\/p>\n\n\n\n

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

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

So \u03c1 is non-negative only if<\/p>\n\n\n\n

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

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

\u2264 1, (5.22)<\/p>\n\n\n\n

with the equality holding for<\/p>\n\n\n\n

pure states: |~p| = 1; det \u03c1 = 0. (5.23)<\/p>\n\n\n\n

The vector ~p, also referred to as the polarization vector, is a point on or<\/p>\n\n\n\n

inside the unit sphere: the Bloch ball. Thus, states of single qubits can be<\/p>\n\n\n\n

represented on the Bloch sphere if they are pure and inside the Bloch sphere<\/p>\n\n\n\n

if they are mixed.<\/p>\n\n\n\n

FIGURE 5.2: Bloch ball: points inside the Bloch sphere represent qubits in<\/p>\n\n\n\n

mixed states<\/p>\n\n\n\n

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

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

Example 5.1.6. For a pure state, the Bloch representation of the density<\/p>\n\n\n\n

matrix is of the form<\/p>\n\n\n\n

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

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

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

( + \u02c6p \u00b7 ~\u03c3),<\/p>\n\n\n\n

where \u02c6p is the unit polarization vector of the state. To see this, use the Bloch<\/p>\n\n\n\n

sphere representation of the state vector along the direction \u02c6p = {\u03b8, \u03c6}:<\/p>\n\n\n\n

\u03c1(\u02c6p) = |\u03c8( \u02c6p)ih\u03c8(\u02c6p)|<\/p>\n\n\n\n

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

“<\/p>\n\n\n\n

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

\u03b8<\/p>\n\n\n\n

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

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

i\u03c6<\/p>\n\n\n\n

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

\u03b8<\/p>\n\n\n\n

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

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

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

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

\u03b8<\/p>\n\n\n\n

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

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

\u2212i\u03c6<\/p>\n\n\n\n

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

\u03b8<\/p>\n\n\n\n

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

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

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

“<\/p>\n\n\n\n

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

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

\u03b8<\/p>\n\n\n\n

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

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

\u2212i\u03c6<\/p>\n\n\n\n

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

\u03b8<\/p>\n\n\n\n

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

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

\u03b8<\/p>\n\n\n\n

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

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

i\u03c6<\/p>\n\n\n\n

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

\u03b8<\/p>\n\n\n\n

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

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

\u03b8<\/p>\n\n\n\n

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

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

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

\u03b8<\/p>\n\n\n\n

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

#<\/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

“<\/p>\n\n\n\n

cos \u03b8 e<\/p>\n\n\n\n

\u2212i\u03c6<\/p>\n\n\n\n

sin \u03b8<\/p>\n\n\n\n

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

i\u03c6<\/p>\n\n\n\n

sin \u03b8 \u2212cos \u03b8<\/p>\n\n\n\n

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

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

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

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

( + \u02c6p \u00b7 ~\u03c3).<\/p>\n\n\n\n

Exercise 5.6. Calculate the expectation value h\u02c6n \u00b7 ~\u03c3i of the spin along the di-<\/p>\n\n\n\n

rection \u02c6n, in the mixed state characterized by a polarization vector ~p to<\/p>\n\n\n\n

validate the interpretation of ~p as the polarization along the direction \u02c6n.<\/p>\n\n\n\n

Exercise 5.7. Locate in the Bloch ball the states given by the following density<\/p>\n\n\n\n

matrices: (a)<\/p>\n\n\n\n

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

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

“<\/p>\n\n\n\n

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

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

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

(b)<\/p>\n\n\n\n

“<\/p>\n\n\n\n

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

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

#<\/p>\n","protected":false},"excerpt":{"rendered":"

The representation of a single qubit state on the Bloch sphere can be ex- tended to the density operator. The Bloch sphere is parametrized by spherical angles or in terms of the Bloch vector of Equation 4.2, which characterizes the polarization of the state. Can we use this kind of description for a mixed state? […]<\/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-4029","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\/4029","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=4029"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4029\/revisions"}],"predecessor-version":[{"id":4030,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4029\/revisions\/4030"}],"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=4029"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4029"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4029"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}