ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

1.3. Theory of bipartite entanglement 17
1.3.2. Mixed states: entanglement vs separability
A mixed state ϱ can be decomposed as a convex combination of pure states,
ϱ =
pk|ψk〉〈ψk| . (1.28)
k
Eq. (1.28) tells us how to create the state described by the density matrix ϱ: we
have to prepare the state |ψ1〉 with probability p1, the state |ψ2〉 with probability
p2, etc. For instance, we could collect N copies (N ≫ 1) of the system, prepare
nk Npk of them in the state |ψk〉, and pick a random system.
The problem is that the decomposition of Eq. (1.28) is not unique: unless
ϱ is already a pure state, there exist infinitely many decompositions of a generic
ϱ in ensembles of pure states, meaning that the mixed state can be prepared in
infinitely many different ways. One can expect that this has some consequence on
the entanglement. Let us suppose we have a bipartite system and we detect, by local
measurements, the presence of correlations between the two subsystems. Given the
ambiguity on the state preparation, we cannot know a priori if those correlations
arose from a quantum interaction between the subsystems (meaning entanglement)
or were induced by means of LOCC (meaning classical correlations). It is thus
clear that a mixed state can be defined separable (classically correlated) if there
exist at least one way of engineering it by LOCC; on the other hand it is entangled
(quantumly correlated) if, among the infinite possible preparation procedures, there
is no one which relies on LOCC only [264].
Definition 2. A mixed quantum state ϱ of a bipartite system, described by the Hilbert
space H = H1 ⊗ H2, is separable if and only if there exist coefficients {pk | pk ≥
0,
k pk = 1}, and states {σk} ∈ H1 and {τk} ∈ H2, such that
ϱ =
pk (σk ⊗ τk) . (1.29)
Otherwise, ϱ is an entangled state.
k
For pure states, the expansion Eq. (1.29) has a single term and we recover
Def. 1, i.e. the only separable pure states are product states. For mixed states, not
only product states (containing zero correlations of any form) but in general any
convex combination of product states is separable. This is obvious as the state of
Eq. (1.29) contains only classical correlations, since it can be prepared by means of
LOCC.
However, Def. 2 is in all respects impractical. Deciding separability according
to the above definition would imply checking all the infinitely many decomposition
of a state ϱ and looking for at least one of the form Eq. (1.29), to conclude that
the state is not entangled. This is clearly impossible. For this reason, several
operational criteria have been developed in order to detect entanglement in mixed
quantum states [144, 43, 143]. Two of them are discussed in the following.
1.3.2.1. Positive Partial Transposition criterion. One of the most powerful results
to date in the context of separability criteria is the Peres–Horodecki condition
[178, 118]. It is based on the operation of partial transposition of the density
matrix of a bipartite system, obtained by performing transposition with respect to
the degrees of freedom of one subsystem only. Peres criterion states that, if a state
ϱs is separable, then its partial transpose ϱ T1
s (with respect e.g. to subsystem S1) is