ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

96 5. Multimode entanglement under symmetry
correlations of the multimode Gaussian state with CM σ. These quantities are,
clearly, the equivalent marginal purities µαeq and µβeq, the global purity µeq and
the equivalent two-mode invariant ∆eq.
Let us remind that, by exploiting Eqs. (2.61,2.62,5.2), the symplectic spectra of
the CMs σαm and σβn may be recovered by means of the local two-mode invariants
µβ, µα, µ β2, µ α2, ∆β2 and ∆α2. The quantities µαeq and µβeq are easily determined
in terms of local invariants alone:
µαeq = 1/ν + α m µβeq = 1/ν +
β n . (5.6)
On the other hand, the determination of µeq and ∆eq require the additional knowledge
of two global symplectic invariants of the CM σ; this should be expected, because
they are susceptible of quantifying the correlations between the two parties.
The natural choices for the global invariants are the global purity µ = 1/ √ Det σ
and the invariant ∆, given by
One has
∆ = MDet α + M(M − 1)Det ε + NDet β
+N(N − 1)Det ζ + 2MNDet γ .
µeq = (ν − α ) M−1 (ν −
β )N−1 µ , (5.7)
∆eq = ∆ − (M − 1)(ν − α ) 2 − (N − 1)(ν −
β )2 . (5.8)
The entanglement between the M-mode and the N-mode subsystems, quantified
by the logarithmic negativity Eq. (3.8) can thus be easily determined, as it is
the case for two-mode states. In particular, the smallest symplectic eigenvalue ˜ν− eq
of the matrix ˜σeq, derived from σeq by partial transposition, fully quantifies the
entanglement between the M-mode and N-mode partitions. Recalling the results
of Sec. 4.2.1, the quantity ˜ν− eq reads
2˜ν 2 − eq = ˜ ∆eq −
with ˜ ∆eq =
2
µ 2 αeq
˜∆ 2 eq − 4
µ 2 , (5.9)
eq
+ 2
µ 2 − ∆eq .
βeq
The logarithmic negativity E αM |β N
N measuring the bipartite entanglement between
the M-mode and N-mode subsystems is then
E αM |β N
N = max [− log ˜ν− eq, 0] . (5.10)
In the case ν +
α M = ν +
β N , corresponding to the following condition
(a + (M − 1)e1)(a + (M − 1)e2) = (b + (N − 1)z1)(b + (N − 1)z2) , (5.11)
on the standard form covariances Eq. (2.60), the equivalent two-mode state is symmetric
and we can determine also the entanglement of formation, using Eq. (4.17).
Let us note that the possibility of exactly determining the entanglement of formation
of a multimode Gaussian state under a M ×N bipartition is a rather remarkable
consequence, even under the symmetry constraints obeyed by the CM σ. Another
relevant fact to point out is that, since both the logarithmic negativity and the
entanglement of formation are decreasing functions of the quantity ˜ν− eq, the two
measures induce the same entanglement hierarchy on such a subset of “equivalently
symmetric” states (i.e. states whose equivalent two-mode CM σeq is symmetric).