ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

38 2. Gaussian states: structural properties
One obtains [GA3]
where
gp(x) =
Tr ϱ p =
N
gp(νi) , (2.36)
i=1
2p (x + 1) p .
− (x − 1) p
A first remarkable consequence of Eq. (2.36) is that
µ(ϱ) = 1
i νi
1
= √ . (2.37)
Det σ
Regardless of the number of modes, the purity of a Gaussian state is fully determined
by the global symplectic invariant Det σ alone, Eq. (2.33). We recall that the
purity is related to the linear entropy SL via Eq. (1.2), which in CV systems simply
becomes SL = 1 − µ. A second consequence of Eq. (2.36) is that, together with
Eqs. (1.13) and (1.15), it allows for the computation of the Von Neumann entropy
SV , Eq. (1.4), of a Gaussian state ϱ, yielding [211]
where
SV (ϱ) =
N
f(νi) , (2.38)
i=1
x + 1 x + 1 x − 1 x − 1
f(x) ≡ log − log . (2.39)
2 2 2 2
Such an expression for the Von Neumann entropy of a Gaussian state was first
explicitly given in Ref. [115]. Notice that SV diverges on infinitely mixed CV
states, while SL is normalized to 1. Let us remark that, clearly, the symplectic
spectrum of single-mode Gaussian states, which consists of only one eigenvalue ν1,
is fully determined by the invariant Det σ = ν 2 1. Therefore, all the entropies Sp’s
(and SV as well) are just increasing functions of Det σ (i.e. of SL) and induce the
same hierarchy of mixedness on the set of one-mode Gaussian states. This is no
longer true for multi-mode states, even for the relevant, simple instance of two-mode
states [GA3], as we will show in the following.
Accordingly, for an arbitrary Gaussian state the mutual information, Eq. (1.16),
quantifying the total (classical and quantum) correlations between two subsystems
[101], can be computed as well. Namely, for a bipartite Gaussian state with global
CM σ A|B, the mutual information yields [115, 211]
I(σ A|B) = SV (σA) + SV (σB) − SV (σ A|B) , (2.40)
where each Von Neumann entropy can be evaluated from the respective symplectic
spectrum using Eq. (2.38).
2.3.2. Comparison between entropic measures
Here we aim to find extremal values of Sp (for p = 2) at fixed SL in the general Nmode
Gaussian instance, in order to quantitatively compare the characterization
of mixedness given by the different entropic measures [GA3]. For simplicity, in