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arXiv:2307.07685

Gaussian tripartite entanglement in the simultaneous measurement of position and momentum

J. A. Mendoza-Fierro, L. M. Arévalo Aguilar

Published Jul 15, 2023• 0 citations
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Abstract

In this work, we prove the generation of genuine tripartite continuous-variable entanglement in the unitary dynamics of the simultaneous measurement process of position and momentum observables raised by Arthurs and Kelly, considering a measurement configuration where the system under examination is a rotated, displaced, and squeezed vacuum state. Under these assumptions, the measurement configuration is entirely described by a Gaussian state. Then, through the positive partial transpose criterion (PPT), we certify genuine tripartite entanglement by testing the non-separability of the three $\left(1~ \text{vs}~2\right)$-mode bipartitions of the system. This process allows us to classify the qualitative properties of the entanglement in the category of fully inseparable Gaussian states according to the classification exposed in [Giedke et al., \href{https://link.aps.org/doi/10.1103/PhysRevA.64.052303}{Phys. Rev. A \textbf{64}, 052303 (2001)}]. Besides, we determine the quantitative entanglement properties of the system using the residual tripartite R{é}nyi-2 entanglement as a quantifier measure.

DOI
10.48550/arxiv.2307.07685

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In this work, we prove the generation of genuine tripartite continuous-variable entanglement in the unitary dynamics of the simultaneous measurement process of position and momentum observables raised by Arthurs and Kelly, considering a measurement configuration where the system under examination is a rotated, displaced, and squeezed vacuum state. Under these assumptions, the measurement configuration is entirely described by a Gaussian state. Then, through the positive partial transpose criterion (PPT), we certify genuine tripartite entanglement by testing the non-separability of the three (1 vs 2)\left(1~ \text{vs}~2\right)(1 vs 2)-mode bipartitions of the system. This process allows us to classify the qualitative properties of the entanglement in the category of fully inseparable Gaussian states according to the classification exposed in [Giedke et al., \href{https://link.aps.org/doi/10.1103/PhysRevA.64.052303}{Phys. Rev. A \textbf{64}, 052303 (2001)}]. Besides, we determine the quantitative entanglement properties of the system using the residual tripartite R{é}nyi-2 entanglement as a quantifier measure.
The structural aspects of tripartite entanglement in three-mode Gaussian states of continuous variable systems have been studied in [Adesso G, Serafini A and Illuminati F 2006 Phys. Rev. A {\\bf 73} 032345]. Here we focus our attention on the usefulness of such states in the context of realistic processing of continuous-variable quantum information. We introduce and discuss in detail several examples of pure and mixed three-mode states that stand out for their informational and/or entanglement properties. We then describe practical schemes to engineer such states with linear optics. In particular, we introduce a simple procedure -- based on passive optical elements -- to produce pure three-mode Gaussian states with {\\em arbitrary} entanglement structure (upon availability of an initial single-mode squeezed state). We analyze in detail the properties of distributed entanglement, showing that the promiscuity of entanglement sharing is a feature peculiar to symmetric Gaussian states that survives even in the presence of significant degrees of mixedness and decoherence. Next, we discuss the suitability of the considered tripartite entangled states to the implementation of quantum information and communication protocols with continuous variables. This will lead to a feasible experimental proposal to test the promiscuous sharing of continuous-variable entanglement, in terms of the optimal fidelity of teleportation networks with Gaussian resources. As a byproduct, optimal resource states for various communication protocols (among which asymmetric telecloning) will be exactly determined.
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.
We present a complete analysis of the multipartite entanglement of three-mode Gaussian states of continuous-variable systems. We derive standard forms which characterize the covariance matrix of pure and mixed three-mode Gaussian states up to local unitary operations, showing that the local entropies of pure Gaussian states are bound to fulfill a relationship which is stricter than the general Araki-Lieb inequality. Quantum correlations can be quantified by a proper convex roof extension of the squared logarithmic negativity, the continuous-variable tangle, or contangle. We review and elucidate in detail the proof that in multimode Gaussian states the contangle satisfies a monogamy inequality constraint [G. Adesso and F. Illuminati, New J. Phys8, 15 (2006)]. The residual contangle, emerging from the monogamy inequality, is an entanglement monotone under Gaussian local operations and classical communications and defines a measure of genuine tripartite entanglements. We determine the analytical expression of the residual contangle for arbitrary pure three-mode Gaussian states and study in detail the distribution of quantum correlations in such states. This analysis yields that pure, symmetric states allow for a promiscuous entanglement sharing, having both maximum tripartite entanglement and maximum couplewise entanglement between any pair of modes. We thus name these states GHZ∕W\mathrm{GHZ}∕WGHZ∕W states of continuous-variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the WWW states of three qubits. We finally consider the effect of decoherence on three-mode Gaussian states, studying the decay of the residual contangle. The GHZ∕W\mathrm{GHZ}∕WGHZ∕W states are shown to be maximally robust against losses and thermal noise.
We investigate the genuine multipartite nonlocality of three-mode Gaussian states of continuous variable systems. For pure states, we present a simplified procedure to obtain the maximum violation of the Svetlichny inequality based on displaced parity measurements, and we analyze its interplay with genuine tripartite entanglement measured via Rényi-2 entropy. The maximum Svetlichny violation admits tight upper and lower bounds at fixed tripartite entanglement. For mixed states, no violation is possible when the purity falls below 0.86. We also explore a set of recently derived weaker inequalities for three-way nonlocality, finding violations for all tested pure states. Our results provide a strong signature for the nonclassical and nonlocal nature of Gaussian states despite their positive Wigner function, and lead to precise recipes for its experimental verification.
Triple-photon states generated by three-mode spontaneous parametric down-conversion are the paradigm of unconditional non-Gaussian states, essential assets for quantum advantage. How to fully characterize their non-Gaussian entanglement remains however elusive. We propose here a set of sufficient and necessary conditions for separability of the broad family of spontaneously generated three-mode non-Gaussian states. We further derive state-of-the-art conditions for genuine tripartite non-Gaussian entanglement, the strongest class of entanglement. We apply our criteria to triple-photon states revealing that they are fully inseparable and genuinely entangled in moments of order 3n. Our results establish a systematic framework for characterizing the entanglement of triple-photon states and thus fostering their application in quantum information protocols.