Owing to the importance of entanglement as a resource in quantum in-<\/p>\n\n\n\n
formation processing, it is necessary to construct measures of entanglement<\/p>\n\n\n\n
between two component systems. We saw in Chapter 4 a condition for the<\/p>\n\n\n\n
separability of 2-qubit states. For a generic higher dimensional density matrix<\/p>\n\n\n\n
to be separable, a test known as the positive partial transpose (PPT) condi-<\/p>\n\n\n\n
tion was proposed by Peres [55] and the Horodecki\u2019s [42]. The density matrix<\/p>\n\n\n\n
of the system can be expressed as<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,j,l,m<\/p>\n\n\n\n
p<\/p>\n\n\n\n
ijlm<\/p>\n\n\n\n
|iihj| \u2297 |lihm|. (11.56)<\/p>\n\n\n\n
where |ii, |ji are basis states for system A, while |li, |mi are those of B. The<\/p>\n\n\n\n
partial transpose with respect to system B is obtained by interchanging the<\/p>\n\n\n\n
row and column indices of the second system:<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
T<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u2261<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,j,l,m<\/p>\n\n\n\n
p<\/p>\n\n\n\n
ijlm<\/p>\n\n\n\n
|iihj| \u2297 |mihl|. (11.57)<\/p>\n\n\n\n
For separable states, this operator is positive, i.e., has non-negative eigenvalues<\/p>\n\n\n\n
only. If this operator has a negative eigenvalue then the state represented by<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
is entangled.<\/p>\n\n\n\n
Example 11.4.1. It is easy to see that the partial transpose of a separable<\/p>\n\n\n\n
density operator has no negative eigenvalue:<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
p<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2297 \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
, (11.58)<\/p>\n\n\n\n
Taking partial transpose with respect to B is just taking the transpose of the<\/p>\n\n\n\n
reduced matrix \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
. This action does not alter the eigenvalues of \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
and hence<\/p>\n\n\n\n
those of \u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
, which were non-negative to start with.<\/p>\n\n\n\n
Exercise 11.3. Show that the partial transposes of the density matrices for the<\/p>\n\n\n\n
Bell states have a negative eigenvalue.<\/p>\n\n\n\n
Entanglement has so far only been described qualitatively, and we know of<\/p>\n\n\n\n
the two extremes of separable states and maximally entangled 2-qubit states.<\/p>\n\n\n\n
We\u2019d like to develop measures for entanglement that are more quantitative<\/p>\n\n\n\n
and generic. We expect any entanglement measure E(\u03c1) to have the following<\/p>\n\n\n\n
properties.<\/p>\n\n\n\n
1. For an unentangled state, E(\u03c1) = 0.<\/p>\n\n\n\n
Characterization of Quantum Information 233<\/p>\n\n\n\n
2. Local unitary transformations on the system should leave the entangle-<\/p>\n\n\n\n
ment unchanged.<\/p>\n\n\n\n
3. If non-unitary operations are included (for example measurement), then<\/p>\n\n\n\n
the entanglement cannot increase.<\/p>\n\n\n\n
Many di\ufb00erent entanglement measures have been proposed, useful in dif-<\/p>\n\n\n\n
ferent contexts.<\/p>\n\n\n\n
1. Distance measures between the given state and the \u201cnearest\u201d unentan-<\/p>\n\n\n\n
gled state can be directly used.<\/p>\n\n\n\n
2. Entropy of entanglement: If the system at hand (A) is considered as a<\/p>\n\n\n\n
component of a pure state \u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
, expressed in Schmidt form,<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03bb<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
ihi<\/p>\n\n\n\n
A<\/p>\n\n\n\n
| \u2297 |i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
ihi<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|. (11.59)<\/p>\n\n\n\n
the entropy of the reduced density matrix for A is a measure of its entangle-<\/p>\n\n\n\n
ment with B:<\/p>\n\n\n\n
E(A) = S( Tr<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
) = \u2212<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03bb<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
log|\u03bb<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
, (11.60)<\/p>\n\n\n\n
The entropy for the reduced density matrix of B is also the same. Clearly, if<\/p>\n\n\n\n
the two states were unentangled, then they will be pure states themselves and<\/p>\n\n\n\n
the entropy would be zero. This measure also satis\ufb01es the other two conditions<\/p>\n\n\n\n
above. Thus, an entanglement measure for a pure composite state is the von<\/p>\n\n\n\n
Neumann entropy of any of the reduced density matrices.<\/p>\n\n\n\n
This measure is, however, not applicable for mixed states, since the von<\/p>\n\n\n\n
Neumann entropy of a subsystem can be non-zero even if the states are not<\/p>\n\n\n\n
entangled.<\/p>\n\n\n\n
3. Entanglement of formation: Since entanglement is created when the<\/p>\n\n\n\n
system are prepared, one common measure of entanglement is the entangle-<\/p>\n\n\n\n
ment of formation of the entangled pair. Suppose one is to prepare an ensemble<\/p>\n\n\n\n
of states in a given entangled state \u03c1. In one interpretation, the entanglement<\/p>\n\n\n\n
of formation measures the number of Bell states required to construct this<\/p>\n\n\n\n
state. If \u03c1 is constructed out of a mixture of pure states {\u03c6<\/p>\n\n\n\n
i<\/p>\n\n\n\n
}, we have<\/p>\n\n\n\n
\u03c1 =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
p<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|.<\/p>\n\n\n\n
Each state |\u03c8<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i has its own entropy of entanglement E<\/p>\n\n\n\n
i<\/p>\n\n\n\n
. This decomposition<\/p>\n\n\n\n
is not unique, and we have to choose the minimum out of all possible decom-<\/p>\n\n\n\n
positions to de\ufb01ne the entropy of formation of \u03c1:<\/p>\n\n\n\n
E(\u03c1) = min<\/p>\n\n\n\n
“<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
p<\/p>\n\n\n\n
i<\/p>\n\n\n\n
E<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(|\u03c8<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i)<\/p>\n\n\n\n
#<\/p>\n\n\n\n
. (11.61)<\/p>\n\n\n\n
4. Concurrence: This is a somewhat less intuitive measure of entanglement<\/p>\n\n\n\n
but is widely used and is related to the entanglement of formation discussed<\/p>\n\n\n\n
above. It was \ufb01rst proposed by Wootters in 1998 [75].<\/p>\n\n\n\n
<\/p>\n\n\n\n
234 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
We saw in Chapter 4 that a 2-qubit pure state<\/p>\n\n\n\n
|\u03c8i = \u03b1|00i+ \u03b2|01i + \u03b3|10i + \u03b4|11i, (11.62)<\/p>\n\n\n\n
is separable only if \u03b1\u03b4 = \u03b2\u03b3 (Equation 4.9). The di\ufb00erence |\u03b1\u03b4 \u2212 \u03b2\u03b3| can be<\/p>\n\n\n\n
taken to be a measure of entanglement. One way to obtain this is to consider<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u02dc<\/p>\n\n\n\n
\u03c8i = Y<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 Y<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
i, (11.63)<\/p>\n\n\n\n
C(\u03c8) = |h\u03c8|<\/p>\n\n\n\n
\u02dc<\/p>\n\n\n\n
\u03c8i| (11.64)<\/p>\n\n\n\n
= 2|\u03b1\u03b4 \u2212 \u03b2\u03b3| (11.65)<\/p>\n\n\n\n
This can be extended for a mixed state with density matrix \u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
: de\ufb01ne<\/p>\n\n\n\n
\u02dc\u03c1 =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y<\/p>\n\n\n\n
B<\/p>\n\n\n\n
,<\/p>\n\n\n\n
then concurrence can be de\ufb01ned as<\/p>\n\n\n\n
C(\u03c1) = max(0, \u03bb<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u2212 \u03bb<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2212 \u03bb<\/p>\n\n\n\n
3<\/p>\n\n\n\n
\u2212 \u03bb<\/p>\n\n\n\n
4<\/p>\n\n\n\n
), (11.66)<\/p>\n\n\n\n
where the \u03bb<\/p>\n\n\n\n
i<\/p>\n\n\n\n
are the square roots of the eigenvalues of \u03c1\u02dc\u03c1 in decreasing order.<\/p>\n\n\n\n
For two-qubit systems, it turns out that the entanglement of formation is<\/p>\n\n\n\n
related to the concurrence:<\/p>\n\n\n\n
E(\u03c1) = h<\/p>\n\n\n\n
\ue012<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
1 +<\/p>\n\n\n\n
p<\/p>\n\n\n\n
1 \u2212 C<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue011<\/p>\n\n\n\n
\ue013<\/p>\n\n\n\n
, (11.67)<\/p>\n\n\n\n
where h(x) is the standard entropy of a binary probability distribution:<\/p>\n\n\n\n
h(x) = \u2212x log(x) \u2212 (1 \u2212 x) log(1 \u2212 x).<\/p>\n\n\n\n
These measures have dealt only with bipartite entanglement: entanglement<\/p>\n\n\n\n
between two subsystems. There are many more ideas dealing with entangle-<\/p>\n\n\n\n
ment of mixed states that are not discussed here. Neither is the much more<\/p>\n\n\n\n
complex scenario of multipartite entanglement.<\/p>\n\n\n\n
Problems<\/p>\n\n\n\n
11.1. What is the information carried by a throw of a die with 6 faces? What is<\/p>\n\n\n\n
the information carried by n throws of the same die?<\/p>\n\n\n\n
11.2. An experiment produces photons with a 60% probability of being right cir-<\/p>\n\n\n\n
cularly polarized and 40% of being left circularly polarized. Find the entropy<\/p>\n\n\n\n
(i) in an experiment to test for circular polarization; (ii) in an experiment<\/p>\n\n\n\n
to test for linear polarization.<\/p>\n\n\n\n
Problems 235<\/p>\n\n\n\n
11.3. Derive the mutual information relation of Equation 11.18 if the de\ufb01nition<\/p>\n\n\n\n
is Equation 11.19.<\/p>\n\n\n\n
11.4. Consider a preparation of photons that has 70% probability of producing<\/p>\n\n\n\n
right circular polarization and 30% probability of producing vertical polar-<\/p>\n\n\n\n
ization.<\/p>\n\n\n\n
(a) Construct the density matrix for the prepared photon state and \ufb01nd<\/p>\n\n\n\n
its eigenvalues.<\/p>\n\n\n\n
(b) What is the physical meaning of the eigenvectors of this matrix?<\/p>\n\n\n\n
(c) Find the entropy of this system.<\/p>\n\n\n\n
11.5. Prove that for pure states, \u03c1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= \u03c1 =\u21d2 S(\u03c1) = 0.<\/p>\n\n\n\n
11.6. Prove the Araki\u2013Lieb inequality, Equation 11.46.<\/p>\n\n\n\n
11.7. Prove using the Klein inequality that for a d dimensional system, S(\u03c1) \u2264<\/p>\n\n\n\n
log d.<\/p>\n\n\n\n
11.8. Calculate the concurrence for the Bell state |\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i.<\/p>\n","protected":false},"excerpt":{"rendered":"
Owing to the importance of entanglement as a resource in quantum in- formation processing, it is necessary to construct measures of entanglement between two component systems. We saw in Chapter 4 a condition for the separability of 2-qubit states. For a generic higher dimensional density matrix to be separable, a test known as the positive […]<\/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-4148","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\/4148","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=4148"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4148\/revisions"}],"predecessor-version":[{"id":4579,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4148\/revisions\/4579"}],"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=4148"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4148"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}