ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

6.2. Monogamy of distributed entanglement in N-mode Gaussian states 115
6.2.1. General monogamy constraints and residual entanglement
In the broadest setting we want to investigate whether a monogamy inequality like
Ineq. (6.2) holds in the general case of Gaussian states with an arbitrary number
N of modes. Considering a Gaussian state distributed among N parties (each
owning a single mode), the monogamy constraint on distributed entanglement can
be written as
E Si|(S1...Si−1Si+1...SN
N
)
≥ E Si|Sj (6.17)
where the global system is multipartitioned in subsystems Sk (k = 1, . . ., N), each
owned by a respective party, and E is a proper measure of bipartite entanglement.
The corresponding general monogamy inequality, see Eq. (1.53), is known to hold
for qubit systems [169].
The left-hand side of inequality (6.17) quantifies the bipartite entanglement
between a probe subsystem Si and the remaining subsystems taken as a whole.
The right-hand side quantifies the total bipartite entanglement between Si and
each one of the other subsystems Sj=i in the respective reduced states. The nonnegative
difference between these two entanglements, minimized over all choices of
the probe subsystem, is referred to as the residual multipartite entanglement. It
quantifies the purely quantum correlations that are not encoded in pairwise form,
so it includes all manifestations of genuine K-partite entanglement, involving K
subsystems (modes) at a time, with 2 < K ≤ N. The study of entanglement
sharing and monogamy constraints thus offers a natural framework to interpret
and quantify entanglement in multipartite quantum systems [GA12] (see Sec. 1.4).
To summarize the results we are going to present, it is now known that the
(Gaussian) contangle — and the Gaussian tangle, as an implication — is monogamous
in fully symmetric Gaussian states of N modes [GA10]. In general, we
have proven the Gaussian tangle to be monogamous in all, pure or mixed, Gaussian
states of an arbitrary number of modes [GA15]. A full analytical proof of the
monogamy inequality for the contangle in all Gaussian states beyond the symmetry,
is currently lacking; however, numerical evidence obtained for randomly generated
nonsymmetric 4-mode Gaussian states strongly supports the conjecture that the
monogamy inequality be true for all multimode Gaussian states, using also the
(Gaussian) contangle as a measure of bipartite entanglement [GA10]. Remarkably,
for all (generally nonsymmetric) three-mode Gaussian states the (Gaussian) contangle
has been proven to be monogamous, leading in particular to a proper measure
of tripartite entanglement in terms of the residual contangle: the analysis of distributed
entanglement in the special instance of three-mode Gaussian states, with
all the resulting implications, is postponed to the next Chapter.
j=i
6.2.2. Monogamy inequality for fully symmetric states
The analysis of Sec. 5.2 has revealed that in fully permutation-invariant Gaussian
states, by comparing the bipartite block entanglement in the various bipartitions of
the modes (which is always unitarily localizable into a two-mode one [GA4, GA5]),
the presence of genuine multipartite entanglement is revealed. In general, with
increasing number of modes, we have evidenced by scaling arguments how the
individual pairwise entanglement between any two modes is redistributed into a
multipartite entanglement among all the modes.