![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg72.png\")
<\/p>\n\n\n\nWe now have an alternate formulation of quantum mechanics, in terms<\/p>\n\n\n\n
of density operators instead of state vectors, that is good for open systems<\/p>\n\n\n\n
as well. Let\u2019s go through the axioms of quantum mechanics framed in this<\/p>\n\n\n\n
language.<\/p>\n\n\n\n
5.2.1 States and observables<\/p>\n\n\n\n
Postulate 1. Quantum State: The state of a quantum system is described<\/p>\n\n\n\n
by a density operator in Hilbert space, i.e., a positive Hermitian operator with<\/p>\n\n\n\n
unit trace.<\/p>\n\n\n\n
Postulate 2. Observables: An observable A is represented by a Hermitian<\/p>\n\n\n\n
operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A on Hilbert space. When measured in a state \u03c1, the probability of<\/p>\n\n\n\n
an outcome a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
is given by<\/p>\n\n\n\n
P(a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
)<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
= Tr(\u03c1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
P<\/p>\n\n\n\n
n<\/p>\n\n\n\n
) (5.24)<\/p>\n\n\n\n
where<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
P<\/p>\n\n\n\n
n<\/p>\n\n\n\n
= |nihn| is the projection on the appropriate eigenspace of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A. The<\/p>\n\n\n\n
expectation value of the observable is given by<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Ai<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
= Tr(\u03c1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A). (5.25)<\/p>\n\n\n\n
5.2.2 Generalized measurements<\/p>\n\n\n\n
When measurements are made on open systems, we are forced to generalize<\/p>\n\n\n\n
our notion (from Section 3.3) of projections on the eigenspaces of the observ-<\/p>\n\n\n\n
able being measured. Those are special cases and are called von Neumann or<\/p>\n\n\n\n
projective measurements.<\/p>\n\n\n\n
Most real measurements are not of this kind. To take a simple but extreme<\/p>\n\n\n\n
example: how do we describe the measurement of the position of a photon in<\/p>\n\n\n\n
<\/p>\n\n\n\n
Mixed States, Open Systems, and the Density Operator 89<\/p>\n\n\n\n
an experiment where it strikes a screen that emits a phosphorescent \ufb02ash?<\/p>\n\n\n\n
In this case the position may be noted, but the photon has been absorbed<\/p>\n\n\n\n
by the screen! Thus we can no longer say the measurement is projective with<\/p>\n\n\n\n
the post-measurement state being given by Equation 3.17. In fact the pho-<\/p>\n\n\n\n
ton itself is destroyed by measurement. Another assumption of the projective<\/p>\n\n\n\n
measurement model is that the measurement is repeatable: successive actions<\/p>\n\n\n\n
of the projection operator on the same state give the same result. Most real<\/p>\n\n\n\n
measurements are not repeatable. We therefore need to generalize the idea of<\/p>\n\n\n\n
measurement.<\/p>\n\n\n\n
The main characteristic of any operator representing measurement is that<\/p>\n\n\n\n
it must tell us how to calculate the probabilities of outcomes. The projective<\/p>\n\n\n\n
measurements considered in Chapter 3 can be expressed in the density opera-<\/p>\n\n\n\n
tor formalism as follows. If the outcome is \u03b1 then the state is transformed by<\/p>\n\n\n\n
the projection operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
= |\u03b1ih\u03b1|:<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
Measure<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A, obtain \u03b1<\/p>\n\n\n\n
\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2192<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
. (5.26)<\/p>\n\n\n\n
The probability of obtaining the outcome \u03b1 is given by<\/p>\n\n\n\n
P(\u03b1) = Tr(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
) = Tr(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
\u03c1) = Tr(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
\u03c1). (5.27)<\/p>\n\n\n\n
The last step follows from the orthogonality of projection operators (Equa-<\/p>\n\n\n\n
tion 3.21):<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
. It is this property that we drop in the case of gener-<\/p>\n\n\n\n
alized measurements.<\/p>\n\n\n\n
For generalized measurement, we think in terms of a complete set of mea-<\/p>\n\n\n\n
surement operators<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
, each of which corresponds to a di\ufb00erent measurement<\/p>\n\n\n\n
outcome m. But these operators do not need to be orthogonal like projection<\/p>\n\n\n\n
operators.<\/p>\n\n\n\n
FIGURE 5.4: Generalized measurement.<\/p>\n\n\n\n
Postulate 3. Measurement: a measurement process capable of yielding m<\/p>\n\n\n\n
possible distinct outcomes can be described by a set of Hermitian measurement<\/p>\n\n\n\n
operators<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
satisfying<\/p>\n\n\n\n
P<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
= (the completeness relation). The<\/p>\n\n\n\n
probability of an outcome m is<\/p>\n\n\n\n
P(m) = Tr(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u03c1) (5.28)<\/p>\n\n\n\n
<\/p>\n\n\n\n
90 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
and the state after measurement is given by the density operator<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
m<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
Tr(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u03c1)<\/p>\n\n\n\n
. (5.29)<\/p>\n\n\n\n
The special case of projective measurements corresponds to<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u2261<\/p>\n\n\n\n
|mihm| =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
.<\/p>\n\n\n\n
5.2.3 Measurements of the POVM kind<\/p>\n\n\n\n
In most applications of measurement, we are not interested in the post-<\/p>\n\n\n\n
measurement state of the system, but only in the statistics, or the relative<\/p>\n\n\n\n
probabilities of di\ufb00erent outcomes, that we can collect by measuring an en-<\/p>\n\n\n\n
semble. A special case of the measurement postulate caters to this need, and is<\/p>\n\n\n\n
known as the POVM formalism. The set of measurement operators is known<\/p>\n\n\n\n
as a positive operator-valued measure<\/p>\n\n\n\n
2<\/p>\n\n\n\n
or POVM for short. The reason<\/p>\n\n\n\n
for this technical-sounding name is not important; we will just describe the<\/p>\n\n\n\n
main elements of this formalism here, due to its usefulness and pervasiveness<\/p>\n\n\n\n
in literature.<\/p>\n\n\n\n
If we consider the set of operators<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
= M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
,<\/p>\n\n\n\n
X<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
= , (5.30)<\/p>\n\n\n\n
then the probability of outcome m on making a measurement on the state \u03c1<\/p>\n\n\n\n
is<\/p>\n\n\n\n
P(m) = Tr(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u03c1).<\/p>\n\n\n\n
It can be easily seen that the operators<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
are positive, but not necessarily<\/p>\n\n\n\n
orthogonal. That is,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
n<\/p>\n\n\n\n
6= \u03b4<\/p>\n\n\n\n
mn<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
.<\/p>\n\n\n\n
They are called the POVM elements, with the set {<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
} called the POVM.<\/p>\n\n\n\n
For our purposes, the POVM is just a set of positive operators that add up<\/p>\n\n\n\n
to unity. Some texts also call these operators as forming a non-orthogonal<\/p>\n\n\n\n
partition of unity (as opposed to the orthogonal partition made by projection<\/p>\n\n\n\n
operators).<\/p>\n\n\n\n
Example 5.2.1. If we consider the projectors<\/p>\n\n\n\n
m<\/p>\n\n\n\n
= |mihm| as the measure-<\/p>\n\n\n\n
ment operators, then POVM elements are<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
m<\/p>\n\n\n\n
=<\/p>\n\n\n\n
m<\/p>\n\n\n\n
,<\/p>\n\n\n\n
the same as the measurements operators themselves. Some texts call these<\/p>\n\n\n\n
projection-valued measures or PVMs.<\/p>\n\n\n\n
2<\/p>\n\n\n\n
The word \u201cmeasure\u201d becomes relevant only in the case of in\ufb01nite dimensional Hilbert<\/p>\n\n\n\n
spaces.<\/p>\n\n\n\n
<\/p>\n\n\n\n
Mixed States, Open Systems, and the Density Operator 91<\/p>\n\n\n\n
Example 5.2.2. One context in which POVM is very useful is in distinguishing<\/p>\n\n\n\n
two non-orthogonal states with maximum probability. Consider for example<\/p>\n\n\n\n
the states<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i = |0i, |\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i = |+i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i + |1i).<\/p>\n\n\n\n
The operators |1ih1| and |\u2212ih\u2212| project onto orthogonal subspaces. We can<\/p>\n\n\n\n
form a partition of unity by adding a third operator, so that the set<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
1<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
2 \u2212<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue011<\/p>\n\n\n\n
|1ih1|,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
2<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
2 \u2212<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue011<\/p>\n\n\n\n
|\u2212ih\u2212|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
3<\/p>\n\n\n\n
= \u2212 (<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
2<\/p>\n\n\n\n
).<\/p>\n\n\n\n
forms a POVM. Verify that each of the<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
i<\/p>\n\n\n\n
is positive.<\/p>\n\n\n\n
If we measure these operators,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
1<\/p>\n\n\n\n
and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
2<\/p>\n\n\n\n
giving outcomes yield positive<\/p>\n\n\n\n
conclusions: there will be no outcome corresponding to<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
1<\/p>\n\n\n\n
if the state were<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i, and none corresponding to<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
2<\/p>\n\n\n\n
if the state were |\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i. But when the<\/p>\n\n\n\n
outcome corresponding to<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
3<\/p>\n\n\n\n
occurs, then we cannot tell which state we had.<\/p>\n\n\n\n
These operators thus give us a way of unambiguously distinguishing the two<\/p>\n\n\n\n
states except in the third (inconclusive) case.<\/p>\n\n\n\n
The POVM formalism is especially useful when we consider a system in<\/p>\n\n\n\n
a mixed state as a subspace of a larger system in a pure state. If we per-<\/p>\n\n\n\n
form projective measurements on a larger space, the e\ufb00ect on the subspace<\/p>\n\n\n\n
is of POVM measurements (see Box 5.1). This is in fact the motivation for a<\/p>\n\n\n\n
theorem due to Neumark, which states that any POVM can be realized as a<\/p>\n\n\n\n
projective measurement on an extended Hilbert space.<\/p>\n\n\n\n
5.2.4 State evolution<\/p>\n\n\n\n
How is the evolution of a system described in terms of density matrices?<\/p>\n\n\n\n
The evolution operator U for a closed system must be unitary. So for a closed<\/p>\n\n\n\n
system evolving from initial time t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= 0 to some \ufb01nal time t, we can write<\/p>\n\n\n\n
\u03c1(t) = U(t)\u03c1(0)U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
(t). (5.31)<\/p>\n\n\n\n
For a mixed state, \u03c1 =<\/p>\n\n\n\n
P<\/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
|. Assuming that time evolution pre-<\/p>\n\n\n\n
serves this linearity, we can extend Equation 5.31:<\/p>\n\n\n\n
\u03c1(t) =<\/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
U(t)|\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
|U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
(t). (5.32)<\/p>\n\n\n\n
Now the unitary time-evolution operator is obtained from the energy operator,<\/p>\n\n\n\n
or Hamiltonian<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H for the system:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U(t) = exp(\u2212i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H(t \u2212 t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)\/~)<\/p>\n","protected":false},"excerpt":{"rendered":"
We now have an alternate formulation of quantum mechanics, in terms of density operators instead of state vectors, that is good for open systems as well. Let\u2019s go through the axioms of quantum mechanics framed in this language. 5.2.1 States and observables Postulate 1. Quantum State: The state of a quantum system is described by […]<\/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-4033","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\/4033","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=4033"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4033\/revisions"}],"predecessor-version":[{"id":4569,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4033\/revisions\/4569"}],"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=4033"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4033"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4033"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}