ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

142 7. Tripartite entanglement in three-mode Gaussian states
the same rate (different γi) or under the same amount of thermal photons (different
ni), then the obvious, optimal way to shield tripartite entanglement is concentrating
it, by unitary localization, in the two least decoherent modes. The entanglement
can then be redistributed among the three modes by a reversal unitary operation,
just before employing the state. Of course, the concentration and distribution of
entanglement require a high degree of non-local control on two of the three-modes,
which would not always be allowed in realistic operating conditions.
As a final remark, let us mention that the bipartite entanglement of GHZ/W
states under 1 × 2 bipartitions, decays slightly faster (in homogeneous baths with
equal number of photons) than that of an initial pure two-mode squeezed state with
the same initial entanglement. In this respect, the multimode entanglement is more
fragile than the two-mode one, as the Hilbert space exposed to decoherence which
contains it is larger.
7.4.2. Entanglement distribution in noisy GHZ/W states
We consider here the noisy version of the GHZ/W states previously introduced
(Sec. 7.3.1), which are a family of mixed Gaussian fully symmetric states, also
called three-mode squeezed thermal states [53]. They result in general from the
dissipative evolution of pure GHZ/W states in proper Gaussian noisy channels, as
shown in Sec. 7.4.1. Let us mention that various properties of noisy three-mode
Gaussian states have already been addressed, mainly regarding their effectiveness
in the implementation of CV protocols [184, 84]. Here, based on Ref. [GA16], we
focus on the multipartite entanglement properties of noisy states. This analysis
will allow us to go beyond the set of pure states, thus gaining deeper insight into
the role played by realistic quantum noise in the sharing and characterization of
tripartite entanglement.
Noisy GHZ/W states are described by a CM σ th
s of the form Eq. (2.60), with
α = a2, ε = diag{e + , e − } and
e ± = a2 − n 2 ± (a 2 − n 2 ) (9a 2 − n 2 )
, (7.52)
4a
where a ≥ n to ensure the physicality of the state. Noisy GHZ/W states have a
completely degenerate symplectic spectrum (their symplectic eigenvalues being all
equal to n) and represent thus, somehow, the three-mode generalization of twomode
squeezed thermal states (also known as symmetric GMEMS, states of maximal
negativity at fixed purities, see Sec. 4.3.3.1). The state σth s is completely
determined by the local purity µl = a−1 and by the global purity µ = n−3 . Noisy
GHZ/W states reduce to pure GHZ/W states (i.e. three-mode squeezed vacuum
states) for n = 1.
For ease of notation, let us replace the parameter a with the effective “squeezing
degree” s, defined by
s = 1
3
2
3a2 + √ 9a4 − 10n2a2 + n4 n2 − 5 , (7.53)
whose physical significance will become clear once the optical state engineering of
noisy GHZ/W will be described in Sec. 10.1.2.2.
7.4.2.1. Separability properties. Depending on the defining parameters s and n, noisy
GHZ/W states can belong to three different separability classes [94] according to the