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arXiv:0804.4703 · Physical Review Letters

Quantum Nonlocality and Partial Transposition for Continuous-Variable Systems

Alejo Salles, Daniel Cavalcanti, Antonio Acín

Published Jul 25, 2008• 25 citations
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Abstract

A continuous-variable Bell inequality, valid for an arbitrary number of observers measuring observables with an arbitrary number of outcomes, was recently introduced [Cavalcanti, Phys. Rev. Lett. 99, 210405 (2007)10.1103/PhysRevLett.99.210405]. We prove that any n-mode quantum state violating this inequality with quadrature measurements necessarily has a negative partial transposition. Our results thus establish the first link between nonlocality and bound entanglement for continuous-variable systems.

DOI
10.1103/physrevlett.101.040404

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A continuous-variable Bell inequality, valid for an arbitrary number of observers measuring observables with an arbitrary number of outcomes, was recently introduced [Cavalcanti, Phys. Rev. Lett. 99, 210405 (2007)10.1103/PhysRevLett.99.210405]. We prove that any n-mode quantum state violating this inequality with quadrature measurements necessarily has a negative partial transposition. Our results thus establish the first link between nonlocality and bound entanglement for continuous-variable systems.
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.
A Bell inequality is a fundamental test to rule out local hidden variable model descriptions of correlations between two physically separated systems. There have been a number of experiments in which a Bell inequality has been violated using discrete-variable systems. We demonstrate a violation of Bell's inequality using continuous variable quadrature measurements. By creating a four-mode entangled state with homodyne detection, we recorded a clear violation with a Bell value of B=2.31±0.02. This opens new possibilities for using continuous variable states for device independent quantum protocols.
There have been theoretical and experimental studies on quantum nonlocality for continuous variables, based on dichotomic observables. In particular, we are interested in two cases of dichotomic observables for the light field of continuous variables: One case is even and odd numbers of photons and the other case is no photon and the presence of photons. We analyze various observables to give the maximum violation of Bell's inequalities for continuous-variable states. We discuss an observable which gives the violation of Bell's inequality for any entangled pure continuous-variable state. However, it does not have to be a maximally entangled state to give the maximal violation of Bell's inequality. This is attributed to a generic problem of testing the quantum nonlocality of an infinite-dimensional state using a dichotomic observable.
We investigate the violation of noncontextuality by a class of continuous-variable states, including variations of entangled coherent states and a two-mode continuous superposition of coherent states. We generalize the Kochen-Specker (KS) inequality discussed by Cabello [A. Cabello, Phys. Rev. Lett. 101, 210401 (2008)] by using effective bidimensional observables implemented through physical operations acting on continuous-variable states, in a way similar to an approach to the falsification of Bell-Clauser-Horne-Shimony-Holt inequalities put forward recently. We test for state-independent violation of KS inequalities under variable degrees of state entanglement and mixedness. We then demonstrate theoretically the violation of a KS inequality for any two-mode state by using pseudospin observables and a generalized quasiprobability function.