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Physical Review A

Spatiotemporal multipartite entanglement

Mikhail I. Kolobov, Giuseppe Patera

Published May 11, 2011• 9 citations
Publisher

Abstract

In this Rapid Communication, we propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous-variable Gaussian states by considering, instead of the global covariance matrix, the local correlation matrix at two different spatiotemporal points $(\mathrm{\ensuremath{\rho}P\vec},t)$ and $(\mathrm{\ensuremath{\rho}P\vec} {}^{'},{t}^{'})$, with $\mathrm{\ensuremath{\rho}P\vec}$ being the transverse coordinate. Our approach makes it possible to introduce the characteristic spatial length and the characteristic time of the multipartite entanglement, which in general depend on the number of ``parties'' in the system. As an example, we consider tripartite entanglement in spontaneous parametric down-conversion with a spatially structured pump. We investigate spatiotemporal properties of such entanglement and calculate its characteristic spatial length and time.

DOI
10.1103/physreva.83.050302

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In this Rapid Communication, we propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous-variable Gaussian states by considering, instead of the global covariance matrix, the local correlation matrix at two different spatiotemporal points (\mathrm{\ensuremath{\rho}P\vec},t) and (\mathrm{\ensuremath{\rho}P\vec} {}^{'},{t}^{'}), with \mathrm{\ensuremath{\rho}P\vec} being the transverse coordinate. Our approach makes it possible to introduce the characteristic spatial length and the characteristic time of the multipartite entanglement, which in general depend on the number of ``parties'' in the system. As an example, we consider tripartite entanglement in spontaneous parametric down-conversion with a spatially structured pump. We investigate spatiotemporal properties of such entanglement and calculate its characteristic spatial length and time.
Quantum multipartite entanglement is a striking phenomenon predicted by quantum mechanics when several parts of a physical system share the same quantum state that cannot be factorized into the states of individual subsystems. The Gaussian quantum states are usually characterized by the covariance matrix of the quadrature components. A powerful formalism for treating the Gaussian states is that of the symplectic eigenvalues. In particular, a quantitative measure of multipartite entanglement is the so-called logarithmic negativity, related to the symplectic eigenvalues of the partially transposed covariance matrix. Considering only global variances of the field quadratures one completely neglects the spatiotemporal properties of the electromagnetic field. We propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous variable Gaussian states by considering the local correlation matrix at two different spatiotemporal points [see manuscript for characters] and [see manuscript for characters] with [see manuscript for characters] being the transverse coordinate. For stationary and homogeneous systems one can also introduce the spatiotemporal Fourier components of the correlation matrix. The formalism of the global symplectic eigenvalues can be straightforwardly generalized to the frequency-dependent symplectic eigenvalues. This generalized theory allows, in particular, to introduce the characteristic spatial area and time of the multipartite entanglement, which in general depend on the number of "parties" in the system. As an example we consider multipartite entanglement in spontaneous parametric down-conversion with spatially-structured pump. We investigate spatial properties of such entanglement and calculate its characteristic spatial length.
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.
While spatial quantum correlations have been studied in great detail, much less is known about the genuine quantum correlations that can be exhibited by temporal processes. Employing the quantum comb formalism, processes in time can be mapped onto quantum states, with the crucial difference that temporal correlations have to satisfy causal ordering, while their spatial counterpart is not constrained in the same way. Here, we exploit this equivalence and use the tools of multipartite entanglement theory to provide a comprehensive picture of the structure of correlations that (causally ordered) temporal quantum processes can display. First, focusing on the case of a process that is probed at two points in time -- which can equivalently be described by a tripartite quantum state -- we provide necessary as well as sufficient conditions for the presence of bipartite entanglement in different splittings. Next, we connect these scenarios to the previously studied concepts of quantum memory, entanglement breaking superchannels, and quantum steering, thus providing both a physical interpretation for entanglement in temporal quantum processes, and a determination of the resources required for its creation. Additionally, we construct explicit examples of W-type and GHZ-type genuinely multipartite entangled two-time processes and prove that genuine multipartite entanglement in temporal processes can be an emergent phenomenon. Finally, we show that genuinely entangled processes across multiple times exist for any number of probing times.
Multipartite entanglement is quite difficult to deal with. In this paper, I will consider a particular kind of tripartite Gaussian states, which can be produced as the initial pure tripartite states undergo different noise for each party. I will make use of the relative entropy of entanglement to evaluate the entanglement properties of the states based on the fact that the separability of tripartite Gaussian states is well known. In addition, the procedure of calculating the relative entropy between Gaussian states have developed recently. All these make the evaluating the relative entropy of entanglement of tripartite Gaussian states possible.
We present a simple model together with its physical implementation which allows one to generate multipartite entanglement between several spatial modes of the electromagnetic field. It is based on parametric down-conversion with NNN pairs of symmetrically tilted plane waves serving as a pump. The characteristics of this spatial entanglement are investigated in the cases of zero as well as nonzero phase mismatch. Furthermore, the phenomenon of entanglement localization in just two spatial modes is studied in detail and shown to result in an enhancement of the entanglement by a factor N\sqrt{N}N​.