ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
154 8. Unlimited promiscuity of multipartite Gaussian entanglement increasing a. 24 Moreover, as previously shown, the two pairs can themselves be arbitrarily entangled with each other with increasing squeezing s. By constructing a simple and feasible example we have shown that, when the quantum correlations arise among degrees of freedom spanning an infinitedimensional space of states (characterized by unbounded mean energy), an accordingly infinite freedom is tolerated for quantum information to be doled out. This happens with no violation of the fundamental monogamy constraint that retains its general validity in quantum mechanics. In the CV instance demonstrated here, the only effect of monogamy is to bound the divergence rates of the individual entanglement contributions as the squeezing parameters are increased. Within the restricted Hilbert space of four or more qubits, instead, an analogous entanglement structure between the single qubits is strictly forbidden. Quite naturally, the procedure presented here can be in principle extended to investigate the increasingly richer structure of entanglement sharing in N-mode (N even) Gaussian states, via additional pairs of redistributing two-mode squeezing operations. In summary, the main result of this Chapter may be stated as follows [GA19]. ➢ Unlimited promiscuity of multipartite Gaussian entanglement. The entanglement in N-mode Gaussian states (N ≥ 4) can distribute in such a way that it approaches infinity both as a genuinely multipartite quantum correlation shared among all modes, and as a bipartite, two-mode quantum correlation in different pairs of modes, within the validity of the general monogamy constraints on entanglement sharing. 8.2.4. Discussion The discovery of an unlimited promiscuous entanglement sharing, while of inherent importance in view of novel implementations of CV systems for multiparty quantum information protocols, opens unexplored perspectives for the understanding and characterization of entanglement in multiparticle systems. Gaussian states with finite squeezing (finite mean energy) are somehow analogous to discrete systems with an effective dimension related to the squeezing degree [40]. As the promiscuous entanglement sharing arises in Gaussian states by asymptotically increasing the squeezing to infinity, it is natural to expect that dimensiondependent families of states will exhibit an entanglement structure gradually more promiscuous with increasing Hilbert space dimension towards the CV limit. A proper investigation into the huge moat of qudits (with dimension 2 < d < ∞) is therefore the next step to pursue, in order to aim at developing the complete picture of entanglement sharing in many-body systems, which is currently lacking (see Sec. 1.4). Here [GA19], we have initiated this program by establishing a sharp discrepancy between the two extrema in the ladder of Hilbert space dimensions: namely, entanglement of CV systems in the limit of infinite mean energy has been proven infinitely more shareable than that of individual qubits. 24 The notion of unlimited entanglement has to be interpreted in the usual asymptotic sense. Namely, given an arbitrarily large entanglement threshold, one can always pick a state in the considered family with squeezing high enough so that its entanglement exceeds the threshold.
8.2. Entanglement in partially symmetric four-mode Gaussian states 155 Once a more comprehensive understanding will be available of the distributed entanglement structure in high-dimensional and CV systems (also beyond the special class of Gaussian states), the task of devising new protocols to translate such potential into full-power practical implementations for what concerns encoding, processing and distribution of shared quantum information, can be addressed as well. We will briefly discuss the optical generation and exploitation of promiscuous entanglement in four-mode Gaussian states in Sec. 10.2.
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8.2. Entanglement in partially symmetric four-mode Gaussian states 155<br />
Once a more comprehensive understanding will be available of the distributed<br />
entanglement structure in high-dimensional and CV systems (also beyond the special<br />
class of Gaussian states), the task of devising new protocols to translate such<br />
potential into full-power practical implementations for what concerns encoding,<br />
processing and distribution of shared quantum information, can be addressed as<br />
well.<br />
We will briefly discuss the optical generation and exploitation of promiscuous<br />
entanglement in four-mode Gaussian states in Sec. 10.2.
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