ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

4.3. Entanglement versus Entropic measures 65
In particular, any GMEMS can be written as an entangled two-mode squeezed
thermal state [satisfying Ineq. (4.30)]. This provides a characterization of twomode
thermal squeezed states as maximally entangled states for given global and
marginal purities. We can restate this result as follows: given an initial tensor
product of (generally different) thermal states, the unitary operation providing the
maximal entanglement for given values of the local purities µi’s is given by a twomode
squeezing, with squeezing parameter determined by Eq. (4.35). Note that the
same states have also been proven to be maximally entangled at fixed global purity
and mean energy [153], as already mentioned. Nonsymmetric two-mode thermal
squeezed states turn out to be separable in the range
µ1µ2
µ ≤
. (4.37)
µ1 + µ2 − µ1µ2
In such a separable region in the space of purities, no entanglement can occur for
states of the form Eq. (4.34), while, outside this region, they are properly GMEMS.
As a consequence, we obtain a sufficient entropic condition for separability: all
two-mode Gaussian states whose purities fall in the separable region defined by
inequality (4.37), are separable.
GLEMS.— We now consider the states that saturate the upper bound in Eq. (4.10).
They determine the class of Gaussian least entangled states for given global and
local purities (GLEMS). Violation of inequality (4.37) implies that
1 + 1
µ 2 ≤ (µ1 + µ2) 2
µ 2 1 µ2 −
2
2
µ .
Therefore, outside the separable region, GLEMS fulfill
∆ = 1 + 1
µ 2 . (4.38)
Considering the symplectic diagonalization of Gaussian states and the definition of
the seralian ∆ = ν2 − + ν2 +, Eq. (2.34), it immediately follows that the Sp (4,R) invariant
condition (4.38) is fulfilled if and only if the symplectic spectrum of the state
takes the form ν− = 1, ν+ = 1/µ. We thus find that GLEMS are characterized by
a peculiar spectrum, with all the mixedness concentrated in one ‘decoupled’ quadrature.
Moreover, by comparing Eq. (4.38) with the uncertainty relation (2.35), it
follows that GLEMS are the mixed Gaussian states of partial minimum uncertainty
(see Sec. 2.2.2.2). They are therefore the most “classical” mixed Gaussian states
and, in a sense, this is compatible with their property of having minimum entanglement
at fixed purities. GLEMS are determined by the standard form correlation
coefficients
c± = 1
µ1µ2 −
4
4
µ 2 +
1 + 1
µ 2 − (µ1 − µ2) 2
µ 2 1 µ2
2
2
± 1
(1 + µ
−4µ1µ2 +
4µ
2 ) µ 2 1 µ22 − µ2 (µ1 + µ2) 22 µ 2 µ 3 1 µ3 2
.
(4.39)
Quite remarkably, recalling the analysis presented in Sec. 2.3.2, it turns out that
the GLEMS at fixed global and marginal purities are also states of minimal global
p−entropy for p < 2, and of maximal global p−entropy for p > 2.