ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

18 1. Characterizing entanglement
a valid density matrix, in particular positive semidefinite, ϱ T1
s ≥ 0. The same holds
naturally for ϱ T2
s . Positivity of the partial transpose (PPT) is therefore a necessary
condition for separability [178]. The converse (i.e. ϱ T1 ≥ 0 ⇒ ϱ separable) is in
general false. The Horodecki’s have proven that it is indeed true for low-dimensional
systems, specifically bipartite systems of dimensionality 2 × 2 and 2 × 3, in which
case PPT is equivalent to separability [118]. For higher dimensional systems, PPT
entangled states (with ϱ T1 ≥ 0) have been shown to exist [122]. These states are
known as bound entangled [119] as their entanglement cannot be distilled to obtain
maximally entangled states. The existence of bound entangled (undistillable) states
with negative partial transposition is conjectured as well [71, 68], yet a fully rigorous
analytical proof of this fact is still lacking [1].
Recently, PPT criterion has been revisited in the continuous variable scenario
by Simon [218], who showed how the transposition operation acquires in infinitedimensional
Hilbert spaces an elegant geometric interpretation in terms of time
inversion (mirror reflection of the momentum operator). It follows that the PPT
criterion is again necessary and sufficient for separability in all (1 + N)-mode
Gaussian states of continuous variable systems 3 with respect to 1 × N bipartitions
[218, 70, 265]. We have extended its validity to “bisymmetric” M × N Gaussian
states [GA5], i.e. invariant under local mode permutations in the M-mode and in
the N-mode partitions, as detailed in Sec. 3.1.1.
1.3.2.2. Entanglement witnesses. A state ϱ is entangled if and only if there exists
a Hermitian operator ˆ
W such that Tr ˆW ϱ < 0 and Tr ˆW σ ≥ 0 for any state
σ ∈ D, where D ⊂ H is the convex and compact subset of separable states
[118, 229]. The operator ˆ W is the witness responsible for detecting entanglement in
the state ϱ. According to the Hahn-Banach theorem, given a convex and compact
set D and given ϱ ∈ D, there exists an hyperplane which separates ϱ from D.
Optimal entanglement witnesses induce an hyperplane which is tangent to the set
D [145]. A sharper detection of separability can be achieved by means of nonlinear
entanglement witnesses, curved towards the set D of separable states [105, 125].
Entanglement witnesses are quite powerful tools to distinguish entangled from
separable states, especially in practical contexts. With some preliminary knowledge
about the form of the states one is willing to engineer or implement in a quantum
information processing, one can systematically find entanglement witnesses in terms
of experimentally accessible observables, to have a direct tool to test the presence
of quantum correlations, as demonstrated in the lab [15, 30, 106, 148].
1.3.3. Mixed states: quantifying entanglement
The issue of quantifying bipartite entanglement cannot be considered accomplished
yet. We are assisting to a proliferation of entanglement measures, each motivated
by a special context in which quantum correlations play a central role, and each
accounting for a different, sometimes inequivalent quantification and ordering of
entangled states. Detailed treatments of the topic can be found e.g. in Refs. [43, 117,
188, 54]. In general, some physical requirements any good entanglement measure
E should satisfy are the following.
3 Gaussian states of a N-mode continuous variable system are by definition states whose
characteristic function and quasi-probability distributions are Gaussian on a 2N-dimensional real
phase space. See Chapter 2 for a rigorous definition.