ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

62 4. Two-mode entanglement
class of (nonsymmetric) two-mode squeezed thermal states. Let Û1,2(r), Eq. (2.21),
be the two mode squeezing operator with real squeezing parameter r ≥ 0, and let
ϱ ⊗
νi be a tensor product of thermal states with CM νν∓ = 2ν− ⊕ 2ν+, where ν∓
denotes, as usual, the symplectic spectrum of the state. Then, a nonsymmetric two-
mode squeezed thermal state ξνi,r is defined as ξνi,r = Û(r)ϱ⊗νi
Û † (r), corresponding
to a standard form CM with
a = ν− cosh 2 r + ν+ sinh 2 r , b = ν− sinh 2 r + ν+ cosh 2 r , (4.29)
c± = ± ν− + ν+
sinh 2r .
2
Inserting Eqs. (4.29) into Eq. (4.14) yields the following condition for a two-mode
squeezed thermal state ξνi,r to be entangled
sinh 2 (2r) > (ν2 + − 1)(ν 2 − − 1)
(ν− + ν+) 2 . (4.30)
For simplicity we can consider the symmetric instance (ν− = ν+ = 1/ √ µ) and
compute the logarithmic negativity Eq. (4.15), which takes the expression
EN (r, µ) = −(1/2) log[e −4r /µ] .
Notice how the completely mixed state (µ → 0) is always separable while, for
any µ > 0, we can freely increase the squeezing r to obtain Gaussian states with
arbitrarily large entanglement. For fixed squeezing, as naturally expected, the
entanglement decreases with decreasing degree of purity of the state, analogously
to what happens in discrete-variable MEMS [261].
It is in order to remark that the notion of Gaussian maximally entangled mixed
states acquires significance if also the mean energy is kept fixed [153], in which case
the maximum entanglement is indeed attained by (nonsymmetric) thermal squeezed
states. This is somehow expected given the result we are going to demonstrate,
namely that those states play the role of maximally entangled Gaussian states at
fixed global and local mixednesses (GMEMS) [GA2, GA3].
4.3.2. Entanglement vs Information (II) – Maximal negativities at fixed local
purities
The next step in the analysis is the unveiling of the relation between the entanglement
of a Gaussian state of CV systems and the degrees of information related
to the subsystems. Maximally entangled states for given marginal mixednesses
(MEMMS) had been previously introduced and analyzed in detail in the context
of qubit systems [GA1]. The MEMMS provide a suitable generalization of pure
states, in which the entanglement is completely quantified by the marginal degrees
of mixedness.
For two-mode Gaussian states, it follows from the expression Eq. (4.13) of ˜ν−
that, for fixed marginal purities µ1,2 and seralian ∆, the logarithmic negativity
is strictly increasing with increasing µ. By imposing the saturation of the upper
bound of Eq. (4.9),
µ = µ max (µ1,2) ≡ (µ1µ2)/(µ1µ2 + |µ1 − µ2|) , (4.31)
we determine the most pure states for fixed marginals; moreover, choosing µ =
µ max (µ1,2) immediately implies that the upper and the lower bounds on ∆ of
Eq. (4.10) coincide and ∆ is uniquely determined in terms of µ1,2, ∆ = 1 + 1/µ max .