ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

5.1. Bipartite block entanglement of bisymmetric Gaussian states 95
5.1.1. Symplectic properties of symmetric states
As a preliminary analysis, it is useful to provide a symplectic parametrization for the
standard form coefficients of any two-mode reduced state of a fully symmetric Nmode
CM σβN , Eq. (2.60). Following the discussion in Sec. 2.4.1.1, the coefficients
b, z1, z2 of the standard form are determined by the local, single-mode invariant
Det β ≡ µ −2
β , and by the symplectic invariants Det σβ2 ≡ µ−2
β2 and ∆β2 ≡ ∆(σβ2). Here µβ (µ β2) is the marginal purity of the single-mode (two-mode) reduced states,
while ∆β2 is the remaining seralian invariant, Eq. (2.34), of the two-mode reduced
states. According to Sec. 4.1, this parametrization is provided, in the present
instance, by the following equations
b = 1
µβ
, z1 = µβ
4 (ɛ− − ɛ+) , z2 = µβ
4 (ɛ− + ɛ+) , (5.2)
with ɛ− =
∆2 β2 − 4
µ 2 β2 ,
and ɛ+ =
∆β2 − 4
µ 2 β
2
− 4
µ 2 β2 .
This parametrization has a straightforward interpretation, because µβ and µ β 2
quantify the local mixednesses and ∆ β 2 regulates the entanglement of the twomode
blocks at fixed global and local purities [GA2] (see Sec. 4.3.3).
Moreover, we can connect the symplectic spectrum of σ β N , given by Eq. (2.61),
to the known symplectic invariants. The (N − 1)-times degenerate eigenvalue ν −
β
is independent of N, while ν +
β N can be simply expressed as a function of the single-
mode purity µβ and the symplectic spectrum of the two-mode block with eigenvalues
ν −
β
and ν+
β 2:
(ν +
βN ) 2 N(N − 2)
= −
µ 2 +
β
(N − 1)
2
N(ν +
β2) 2 + (N − 2)(ν −
β )2 . (5.3)
In turn, the two-mode symplectic eigenvalues are determined by the two-mode
invariants by the relation
2(ν ∓
β )2 = ∆ β 2 ∓
∆ 2 β 2 − 4/µ 2 β 2 . (5.4)
The global purity Eq. (2.37) of a fully symmetric multimode Gaussian state is
µ βN ≡
−1/2
Det σβN = (ν −
β )N−1ν +
βN −1 , (5.5)
and, through Eq. (5.3), can be fully determined in terms of the one- and two-mode
parameters alone. Analogous reasonings and expressions hold of course for the fully
symmetric M-mode block with CM σ α M given by Eq. (2.60).
5.1.2. Evaluation of block entanglement in terms of symplectic invariants
We can now efficiently discuss the quantification of the multimode block entanglement
of bisymmetric Gaussian states. Exploiting our results on the symplectic
characterization of two-mode Gaussian states [GA2, GA3] (see Sec. 4.1) we can
select the relevant quantities that, by determining the correlation properties of
the two-mode Gaussian state with CM σeq, also determine the entanglement and