ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

G Τ res
7.4. Promiscuous entanglement versus noise and asymmetry 141
1.2
1
0.8
0.6
0.4
0.2
0 0.05 0.1 0.15 0.2 0.25 0.3
Γt
Figure 7.5. Evolution of the residual Gaussian contangle Gres τ for GHZ/W
states with local mixedness a = 2 (solid curves) and T states with local mixedness
a = 2.8014 (dashed curves). Such states have equal initial residual contangle.
The uppermost curves refer to a homogeneous bath with n = 0 (pure
losses), while the lowermost curves refer to a homogeneous bath with n = 1. As
apparent, thermal photons are responsible for the vanishing of entanglement
at finite times.
states were found by further numerical inspection. Quite remarkably, the promiscuous
sharing of quantum correlations, proper to GHZ/W states, appears to better
preserve genuine multipartite entanglement against the action of decoherence.
Notice also that, for a homogeneous bath and for all fully symmetric and bisymmetric
three-mode states, the decoherence of the global bipartite entanglement of
the state is the same as that of the corresponding equivalent two-mode states (obtained
through unitary localization, see Fig. 5.1). Indeed, for any bisymmetric state
which can be localized by an orthogonal transformation (like a beam-splitter), the
unitary localization and the action of the decoherent map of Eq. (7.51) commute,
because σ∞ ∝ is obviously preserved under orthogonal transformations (note
that the bisymmetry of the state is maintained through the channel, due to the
symmetry of the latter). In such cases, the decoherence of the bipartite entanglement
of the original three-mode state (with genuine tripartite correlations) is
exactly equivalent to that of the corresponding initial two-mode state obtained
by unitary localization. This equivalence breaks down, even for GHZ/W states
which can be localized through an (orthogonal) beam-splitter transformation, for
non homogeneous baths, i.e. if the thermal photon numbers ni related to different
modes are different — which is the case for different temperatures Ti or for different
frequencies ωi, according to Eq. (7.48) — or if the couplings γi are different. In
this instance let us remark that the unitary localization could provide a way to
cope with decoherence, limiting its hindering effect on entanglement. In fact, let
us suppose that a given amount of genuine tripartite entanglement is stored in a
symmetric (unitarily localizable) three-mode state and is meant to be exploited,
at some (later) time, to implement tripartite protocols. During the period going
from its creation to its actual use such an entanglement decays under the action of
decoherence. Suppose the three modes involved in the process do not decay with