ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

7.3. Sharing structure of tripartite entanglement: promiscuous Gaussian states 137
G Τ res
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3
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1
0
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a
Figure 7.4. Plot, as a function of the single-mode mixedness a, of the tripartite
residual Gaussian contangle Gres τ Eq. (7.40) in the CV GHZ/W states
(dashed red line); in the T states Eq. (7.44) (solid blue line); and in 50 000
randomly generated mixed symmetric three-mode Gaussian states of the form
Eq. (2.60) (dots). The GHZ/W states, that maximize any bipartite entanglement,
also achieve maximal genuine tripartite quantum correlations, showing
that CV entanglement distributes in a promiscuous way in symmetric Gaussian
states. Notice also how all random mixed states have a nonnegative residual
Gaussian contangle. This confirms the results presented in Ref. [GA10], and
discussed in detail and extended in Sec. 7.2.1, on the strict validity of the
CKW monogamy inequality for CV entanglement in three-mode Gaussian
states.
7.3.3. Promiscuous continuous-variable entanglement sharing
The above results, pictorially illustrated in Fig. 7.4, lead to the conclusion that in
symmetric three-mode Gaussian states, when there is no bipartite entanglement in
the two-mode reduced states (like in T states) the genuine tripartite entanglement
is not enhanced, but frustrated. More than that, if there are maximal quantum
correlations in a three-party relation, like in GHZ/W states, then the two-mode
reduced states of any pair of modes are maximally entangled mixed states.
These findings, unveiling a major difference between discrete-variable (mainly
qubits) and continuous-variable systems, establish the promiscuous nature of CV
entanglement sharing in symmetric Gaussian states [GA10]. Being associated with
degrees of freedom with continuous spectra, states of CV systems need not saturate
the CKW inequality to achieve maximum couplewise correlations, as it was instead
the case for W states of qubits, Eq. (1.52). In fact, the following holds.
➢ Promiscuous entanglement in continuous-variable GHZ/W three-mode
Gaussian states. Without violating the monogamy constraint Ineq. (6.2),
pure symmetric three-mode Gaussian states are maximally three-way entangled
and, at the same time, possess the maximum possible entanglement
between any pair of modes in the corresponding two-mode reduced states.
The two entanglements are mutually enhanced.