ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

3.2. How to quantify bipartite Gaussian entanglement 53
Physicality Separability
density matrix ϱ ≥ 0 ϱ TA ≥ 0
covariance matrix σ + iΩ ≥ 0 ˜σ + iΩ ≥ 0
symplectic spectrum νk ≥ 1 ˜νk ≥ 1
Table 3.I. Schematic comparison between the existence conditions and the
separability conditions for Gaussian states, as expressed in different representations.
To be precise, the second column qualifies the PPT condition, which
is always implied by the separability, and equivalent to it in general 1 × N and
bisymmetric M × N Gaussian states.
We can, without loss of generality, rearrange the modes of a N-mode state such
that the corresponding symplectic eigenvalues of the partial transpose ˜σ are sorted
in ascending order
˜ν− ≡ ˜ν1 ≤ ˜ν2 ≤ . . . ≤ ˜νN−1 ≤ ˜νN ≡ ˜ν+ ,
in analogy to what done in Sec. 2.2.2.2 for the spectrum of σ. With this notation,
PPT criterion across an arbitrary bipartition reduces to ˜ν− ≥ 1 for all separable
Gaussian states. As soon as ˜ν− < 1, the corresponding Gaussian state σ is definitely
entangled. The symplectic characterization of physical versus PPT Gaussian states
is summarized in Table 3.I.
3.1.2. Additional separability criteria
Let us briefly mention alternative characterizations of separability for Gaussian
states.
For a general Gaussian state of any NA × NB bipartition, a necessary and
sufficient condition states that a CM σ corresponds to a separable state if and
only if there exists a pair of CMs σA and σB, relative to the subsystems SA and
SB respectively, such that the following inequality holds [265], σ ≥ σA ⊕ σB.
This criterion is not very useful in practice. Alternatively, one can introduce an
operational criterion based on iterative applications of a nonlinear map, that is
independent of (and strictly stronger than) the PPT condition, and completely
qualifies separability for all bipartite Gaussian states [93].
Note also that a comprehensive characterization of linear and nonlinear entanglement
witnesses (see Sec. 1.3.2.2) is available for CV systems [125], as well
as operational criteria (useful in experimental settings) based on the violation of
inequalities involving combinations of variances of canonical operators, for both
two-mode [70] and multimode settings [240].
However, the range of results collected in this Dissertation deal with classes of
bipartite and multipartite Gaussian states in which PPT holds as a necessary and
sufficient condition for separability, therefore it will be our preferred tool to check
for the presence of entanglement in the states under consideration.
3.2. How to quantify bipartite Gaussian entanglement
3.2.1. Negativities
From a quantitative point of view, a family of entanglement measures which are
computable for general Gaussian states is provided by the negativities. Both the