  Quantum mechanics imposes 'monogamy' constraints on the sharing of
entanglement. We show that, despite these limitations, entanglement can be
fully 'promiscuous', i.e. simultaneously present in unlimited two-body and
many-body forms in states living in an infinite-dimensional Hilbert space.
Monogamy just bounds the divergence rate of the various entanglement
contributions. This is demonstrated in simple families of N-mode (N >= 4)
Gaussian states of light fields or atomic ensembles, which therefore enable
infinitely more freedom in the distribution of information, as opposed to
systems of individual qubits. Such a finding is of importance for the
quantification, understanding and potential exploitation of shared quantum
correlations in continuous variable systems. We discuss how promiscuity
gradually arises when considering simple families of discrete variable states,
with increasing Hilbert space dimension towards the continuous variable limit.
Such models are somehow analogous to Gaussian states with asymptotically
diverging, but finite squeezing. In this respect, we find that non-Gaussian
states (which in general are more entangled than Gaussian states), exhibit also
the interesting feature that their entanglement is more shareable: in the
non-Gaussian multipartite arena, unlimited promiscuity can be already achieved
among three entangled parties, while this is impossible for Gaussian, even
infinitely squeezed states.
