ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

64 4. Two-mode entanglement
purities and difference between them. We now wish to exploit the joint information
about global and marginal degrees of purity to achieve a significative characterization
of entanglement, both qualitatively and quantitatively. Let us first investigate
the role played by the seralian ∆ in the characterization of the properties of twomode
Gaussian states. To this aim, we analyze the dependence of the eigenvalue
˜ν− on ∆, for fixed µ1,2 and µ:
∂ ˜ν 2 −
∂ ∆
µ1, µ2, µ
= 1
⎛
⎝
2
˜∆
˜∆ 2 − 1
4µ 2
⎞
− 1⎠
> 0 . (4.33)
The smallest symplectic eigenvalue of the partially transposed state ˜σ is strictly
monotone in ∆. Therefore the entanglement of a generic Gaussian state σ with
given global purity µ and marginal purities µ1,2, strictly increases with decreasing
∆. The seralian ∆ is thus endowed with a direct physical interpretation: given
the global and the two marginal purities, it exactly determines the amount of
entanglement of the state. Moreover, due to inequality (4.10), ∆ is constrained both
by lower and upper bounds; therefore, both maximally and, remarkably, minimally
entangled Gaussian states exist, at fixed global and local degrees of purity. This fact
admirably elucidates the relation between quantum correlations and information in
two-mode Gaussian states [GA2, GA3, GA6], summarized as follows.
➢ Entanglement at given degrees of information encoded in two-mode
Gaussian states. The entanglement, quantified by the negativities, of twomode
(mixed) Gaussian states is tightly bound from above and from below
by functions of the global and the marginal purities, with only one remaining
degree of freedom related to the symplectic invariant ∆.
4.3.3.1. GMEMS and GLEMS: Extremally entangled states and purity-based separability
criteria. We now aim to characterize extremal (maximally and minimally)
entangled Gaussian states for fixed global and marginal purities, along the lines of
[GA2, GA3]. As it is clear from Eq. (4.5), the standard form of states with fixed
marginal purities always satisfies a = 1/µ1, b = 1/µ2. Therefore the complete characterization
of maximally and minimally entangled states is achieved by specifying
the expression of their coefficients c∓.
GMEMS.— Let us first consider the states saturating the lower bound in Eq. (4.10),
which entails maximal entanglement. They are Gaussian maximally entangled
states for fixed global and local purities (GMEMS), admitting the following stan-
dard form parametrization
1
c± = ± −
µ1µ2
1
µ . (4.34)
It is easily seen that such states belong to the class of asymmetric two-mode
squeezed thermal states, Eq. (4.29), with squeezing parameter and symplectic spectrum
given by
tanh 2r = 2(µ1µ2 − µ 2 1µ 2 2/µ) 1/2 /(µ1 + µ2) , (4.35)
ν 2 ∓ = 1
µ + (µ1 − µ2) 2
2µ 2 1 µ2 ∓
2
|µ1 − µ2|
2µ1µ2
(µ1 − µ2) 2
µ 2 1 µ2 2
+ 4
µ . (4.36)