ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

1.4. Multipartite entanglement sharing and monogamy constraints 27
increasing ε with increasing d [66]. While this is only a preliminary analysis, it raises
intriguing questions, pushing the interest in entanglement sharing towards infinitedimensional
systems. In fact, if ε saturated to 1 for d → ∞, this would entail the
really counterintuitive result that entanglement could be freely shared in this limit!
We notice that, being the entanglement capacity infinite for d → ∞, ε vanishes
if the maximum couplewise entanglement is not infinite. And this is the case,
because again an infinite shared entanglement between two two-party reductions
would allow perfect 1 → 2 telecloning [238] exploiting Einstein-Podolsky-Rosen
(EPR) [73] correlations, but this is forbidden by quantum mechanics.
Nevertheless, the study of entanglement sharing in continuous variable systems
yields surprising consequences, as we will show in Part III of this Dissertation. We
will indeed define proper infinite-dimensional analogues of the tangle [GA10, GA15],
and establish the general monogamy inequality (1.53) on entanglement sharing for
all N-mode Gaussian states distributed among N parties [GA15]. An original,
possibly “promiscuous” structure of entanglement sharing in Gaussian states with
some symmetry constraints will be also elucidated [GA10, GA11, GA16, GA19].