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Journal of the Optical Society of America B

Detecting entanglement by symplectic uncertainty relations

Alessio Serafini

Published Jan 26, 2007• 21 citations
Publisher

Abstract

A hierarchy of multimode uncertainty relations on the second moments of n pairs of canonical operators is derived in terms of quantities invariant under linear canonical (i.e., "symplectic") transformations. Conditions for the separability of multimode continuous variable states are derived from the uncertainty relations, generalizing the inequalities obtained previously [Phys. Rev. Lett.96, 110402 (2006)] to states with some transposed symplectic eigenvalues equal to 1. Finally, to illustrate the methodology proposed for the detection of continuous variable entanglement, the separability of multimode noisy Greenberger-Horne-Zeilinger-like states is analyzed in detail with the presented techniques, deriving a necessary and sufficient condition for the separability of such states under an "even" bipartition of the modes.

DOI
10.1364/josab.24.000347

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A hierarchy of multimode uncertainty relations on the second moments of n pairs of canonical operators is derived in terms of quantities invariant under linear canonical (i.e., "symplectic") transformations. Conditions for the separability of multimode continuous variable states are derived from the uncertainty relations, generalizing the inequalities obtained previously [Phys. Rev. Lett.96, 110402 (2006)] to states with some transposed symplectic eigenvalues equal to 1. Finally, to illustrate the methodology proposed for the detection of continuous variable entanglement, the separability of multimode noisy Greenberger-Horne-Zeilinger-like states is analyzed in detail with the presented techniques, deriving a necessary and sufficient condition for the separability of such states under an "even" bipartition of the modes.
Genuine entanglement identification of large scale systems is crucial for quantum computation, quantum communication and quantum learning advantage. In contrast to experiments, where noisy intermediate-scale programmable photonic quantum processors have been developed, theoretically very limited results have been achieved for detecting genuine entanglement of continuous variable multipartite systems. We propose a quite general and efficient entanglement detection framework for all kinds of multipartite entanglement of continuous variable systems based on uncertainty relations and the sign matrix technique. Matrix criteria are demonstrated and can be applied to various entanglement depth and k-separability problems of multimode systems. We illustrate the genuine entanglement conditions of continuous variable Greenberger-Horne-Zeilinger states of more than a hundred million modes in a photon loss and noise environment.
We demonstrate how to efficiently derive a broad class of inequalities for entanglement detection in multimode continuous variable systems. The separability conditions are established from partial transposition (PT) in combination with several distinct necessary conditions for a quantum physical state, which include previously established inequalities as special cases. Remarkably, our method enables us to support Peres' conjecture to its full generality within the framework of Cavalcanti-Foster-Reid-Drummond multipartite Bell inequality [Phys. Rev. Lett. 99, 210405 (2007)] that the nonlocality necessarily implies negative PT entangled states.
Currently available separability criteria for continuous-variable states are generally based on the covariance matrix of quadrature operators. The well-known separability criterion of Duan et al. [L. M. Duan et al., Phys. Rev. Lett. 84, 2722 (2000)] and Simon [R. Simon, Phys. Rev. Lett. 84, 2726 (2000)], for example, gives a necessary and sufficient condition for a two-mode Gaussian state to be separable, but leaves many entangled non-Gaussian states undetected. Here we introduce an improvement of this criterion that enables a stronger entanglement detection. The improved condition is based on the knowledge of an additional parameter, namely, the degree of Gaussianity, and exploits a connection with Gaussianity-bounded uncertainty relations [A. Mandilara and N. J. Cerf, Phys. Rev. A 86, 030102 (2012)]. We exhibit families of non-Gaussian entangled states whose entanglement remains undetected by the Duan-Simon criterion.
Certification and quantification of correlations for multipartite states of quantum systems appear to be a central task in quantum information theory. We give here a unitary quantum-mechanical perspective of both entanglement and Einstein-Podolsky-Rosen (EPR) steering of continuous-variable multimode states. This originates in the Heisenberg uncertainty relations for the canonical quadrature operators of the modes. Correlations of two-party (Nvs1)(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)(Nvs1)-mode states are examined by using the variances of a pair of suitable EPR-like observables. It turns out that the uncertainty sum of these nonlocal variables is bounded from below by local uncertainties and is strengthened differently for separable states and for each one-way unsteerable state. The analysis of the minimal properly normalized sums of these variances yields necessary conditions of separability and EPR unsteerability of (Nvs1)(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)(Nvs1)-mode states in both possible ways of steering. When the states and the performed measurements are Gaussian, then these conditions are precisely the previously known criteria of separability and one-way unsteerability.