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 partial transpose (PPT) condi-
tion was proposed by Peres [55] and the Horodecki’s [42]. The density matrix
of the system can be expressed as
ρ
AB
=
X
i,j,l,m
p
ijlm
|iihj| ⊗ |lihm|. (11.56)
where |ii, |ji are basis states for system A, while |li, |mi are those of B. The
partial transpose with respect to system B is obtained by interchanging the
row and column indices of the second system:
ρ
T
B
≡
X
i,j,l,m
p
ijlm
|iihj| ⊗ |mihl|. (11.57)
For separable states, this operator is positive, i.e., has non-negative eigenvalues
only. If this operator has a negative eigenvalue then the state represented by
ρ
AB
is entangled.
Example 11.4.1. It is easy to see that the partial transpose of a separable
density operator has no negative eigenvalue:
ρ
AB
=
X
i
p
i
ρ
A
i
⊗ ρ
B
i
, (11.58)
Taking partial transpose with respect to B is just taking the transpose of the
reduced matrix ρ
B
i
. This action does not alter the eigenvalues of ρ
B
and hence
those of ρ
AB
, which were non-negative to start with.
Exercise 11.3. Show that the partial transposes of the density matrices for the
Bell states have a negative eigenvalue.
Entanglement has so far only been described qualitatively, and we know of
the two extremes of separable states and maximally entangled 2-qubit states.
We’d like to develop measures for entanglement that are more quantitative
and generic. We expect any entanglement measure E(ρ) to have the following
properties.
1. For an unentangled state, E(ρ) = 0.
Characterization of Quantum Information 233
2. Local unitary transformations on the system should leave the entangle-
ment unchanged.
3. If non-unitary operations are included (for example measurement), then
the entanglement cannot increase.
Many different entanglement measures have been proposed, useful in dif-
ferent contexts.
1. Distance measures between the given state and the “nearest” unentan-
gled state can be directly used.
2. Entropy of entanglement: If the system at hand (A) is considered as a
component of a pure state ρ
AB
, expressed in Schmidt form,
ρ
AB
=
X
i
λ
i
|i
A
ihi
A
| ⊗ |i
B
ihi
B
|. (11.59)
the entropy of the reduced density matrix for A is a measure of its entangle-
ment with B:
E(A) = S( Tr
B
ρ
AB
) = −
X
i
|λ
i
|
2
log|λ
i
|
2
, (11.60)
The entropy for the reduced density matrix of B is also the same. Clearly, if
the two states were unentangled, then they will be pure states themselves and
the entropy would be zero. This measure also satisfies the other two conditions
above. Thus, an entanglement measure for a pure composite state is the von
Neumann entropy of any of the reduced density matrices.
This measure is, however, not applicable for mixed states, since the von
Neumann entropy of a subsystem can be non-zero even if the states are not
entangled.
3. Entanglement of formation: Since entanglement is created when the
system are prepared, one common measure of entanglement is the entangle-
ment of formation of the entangled pair. Suppose one is to prepare an ensemble
of states in a given entangled state ρ. In one interpretation, the entanglement
of formation measures the number of Bell states required to construct this
state. If ρ is constructed out of a mixture of pure states {φ
i
}, we have
ρ =
X
i
p
i
|ψ
i
ihψ
i
|.
Each state |ψ
i
i has its own entropy of entanglement E
i
. This decomposition
is not unique, and we have to choose the minimum out of all possible decom-
positions to define the entropy of formation of ρ:
E(ρ) = min
“
X
i
p
i
E
i
(|ψ
i
i)
#
. (11.61)
4. Concurrence: This is a somewhat less intuitive measure of entanglement
but is widely used and is related to the entanglement of formation discussed
above. It was first proposed by Wootters in 1998 [75].
234 Introduction to Quantum Physics and Information Processing
We saw in Chapter 4 that a 2-qubit pure state
|ψi = α|00i+ β|01i + γ|10i + δ|11i, (11.62)
is separable only if αδ = βγ (Equation 4.9). The difference |αδ − βγ| can be
taken to be a measure of entanglement. One way to obtain this is to consider
|
˜
ψi = Y
A
⊗ Y
B
|ψ
∗
i, (11.63)
C(ψ) = |hψ|
˜
ψi| (11.64)
= 2|αδ − βγ| (11.65)
This can be extended for a mixed state with density matrix ρ
AB
: define
˜ρ =
ˆ
Y
A
⊗
ˆ
Y
B
ρ
∗
ˆ
Y
A
⊗
ˆ
Y
B
,
then concurrence can be defined as
C(ρ) = max(0, λ
1
− λ
2
− λ
3
− λ
4
), (11.66)
where the λ
i
are the square roots of the eigenvalues of ρ˜ρ in decreasing order.
For two-qubit systems, it turns out that the entanglement of formation is
related to the concurrence:
E(ρ) = h
1
2
1 +
p
1 − C
2
, (11.67)
where h(x) is the standard entropy of a binary probability distribution:
h(x) = −x log(x) − (1 − x) log(1 − x).
These measures have dealt only with bipartite entanglement: entanglement
between two subsystems. There are many more ideas dealing with entangle-
ment of mixed states that are not discussed here. Neither is the much more
complex scenario of multipartite entanglement.
Problems
11.1. What is the information carried by a throw of a die with 6 faces? What is
the information carried by n throws of the same die?
11.2. An experiment produces photons with a 60% probability of being right cir-
cularly polarized and 40% of being left circularly polarized. Find the entropy
(i) in an experiment to test for circular polarization; (ii) in an experiment
to test for linear polarization.
Problems 235
11.3. Derive the mutual information relation of Equation 11.18 if the definition
is Equation 11.19.
11.4. Consider a preparation of photons that has 70% probability of producing
right circular polarization and 30% probability of producing vertical polar-
ization.
(a) Construct the density matrix for the prepared photon state and find
its eigenvalues.
(b) What is the physical meaning of the eigenvectors of this matrix?
(c) Find the entropy of this system.
11.5. Prove that for pure states, ρ
2
= ρ =⇒ S(ρ) = 0.
11.6. Prove the Araki–Lieb inequality, Equation 11.46.
11.7. Prove using the Klein inequality that for a d dimensional system, S(ρ) ≤
log d.
11.8. Calculate the concurrence for the Bell state |β
11
i.
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