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Journal of Physics A Mathematical and Theoretical

Generalizations of the pseudospin operator to test the Bell inequality for the TMSV state

M M Dorantes, J. L. Lucio M.

Published Jun 26, 2009• 15 citations
Publisher

Abstract

Chen et al (2002 Phys. Rev. Lett. 88 040406) proposed a method to measure the entanglement of a bipartite quantum system involving continuous variables. The approach is based upon the construction of a pseudospin operator. An important difference with the spin case is an infinite degeneracy of the eigenkets associated with the pseudospin. In this paper, we analyze the role played by such a degeneracy in the study of the locality of the two-mode-squeezed vacuum state. We construct other representations for the pseudospin and study the corresponding Bell inequalities.

DOI
10.1088/1751-8113/42/28/285309

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Chen et al (2002 Phys. Rev. Lett. 88 040406) proposed a method to measure the entanglement of a bipartite quantum system involving continuous variables. The approach is based upon the construction of a pseudospin operator. An important difference with the spin case is an infinite degeneracy of the eigenkets associated with the pseudospin. In this paper, we analyze the role played by such a degeneracy in the study of the locality of the two-mode-squeezed vacuum state. We construct other representations for the pseudospin and study the corresponding Bell inequalities.
We present a phase-space formalism to evaluate Bell inequality violations in continuous variable systems. By doing so we can generalize previous analyses (which have dealt only with pure states) to arbitrary mixed states. We leverage these results to analyze the effect of temperature on violations of Bell inequalities in a two-mode squeezed thermal state, which can become useful in tests of local realism in the presence of thermal noise. We also explore the nonmonotonic relationship between the violations of Bell inequalities and the amount of entanglement present in this family of mixed states. Additionally, we discuss the optimal choices of pseudospin operators for states beyond the two-mode squeezed vacuum.
We generalize Bell's inequalities to biparticle systems with continuous quantum variables. This is achieved by introducing the Bell operator in perfect analogy to the usual spin-1/2 systems. It is then demonstrated that two-mode squeezed vacuum states display quantum nonlocality by using the generalized Bell operator. In particular, the original Einstein-Podolsky-Rosen entangled states, which are the limiting case of the two-mode squeezed vacuum states, can maximally violate Bell's inequality due to Clauser, Horne, Shimony and Holt. The experimental aspect of our scheme and nonlocality of arbitrary biparticle entangled pure states of continuous variables are briefly considered.
We generalize Bell's inequalities to biparty systems with continuous quantum variables. This is achieved by introducing the Bell operator in perfect analogy to the usual spin- 1/2 systems. It is then demonstrated that two-mode squeezed vacuum states display quantum nonlocality by using the generalized Bell operator. In particular, the original Einstein-Podolsky-Rosen states, which are the limiting case of the two-mode squeezed vacuum states, can maximally violate Bell's inequality due to Clauser, Horne, Shimony, and Holt. The experimental aspect of our scheme is briefly considered.
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.
Quantum networks enable forms of nonlocality beyond the standard Bell scenario, with a multitude of potential applications. Continuous-variable (CV) platforms are particularly attractive for large-scale networks, offering deterministic entanglement generation and favorable prospects for long-distance distribution. Here, we present a formalism to study CV network nonlocality using pseudospin measurements. Considering the linear chain and star configurations, we derive the maximal violations of the corresponding network locality inequalities for arbitrary two-mode states. Using two-mode squeezed vacuum states, we show that the strength of nonlocality in the star configuration remains independent of the network size. Moreover, the nonlocal correlations persist even at arbitrarily high temperatures provided the squeezing exceeds a critical threshold. Further, we demonstrate non-Gaussianity as an enhancer of network nonlocality through illustrations of various classes of non-Gaussian resources. Remarkably, a coherent superposition of single-photon subtractions across modes achieves maximal violation for arbitrarily small squeezing. Finally, we provide schematics of an experimentally feasible implementation of CV network nonlocality based on the isomorphism between pseudospin and spatial parity observables.
It is a long-standing belief, as pointed out by Bell in 1986, that it is impossible to use a two-mode Gaussian state possessing a positive-definite Wigner function to demonstrate nonlocality as the Wigner function itself provides a local hidden-variable model. In particular, when one performs continuous-variable (CV) quadrature measurements upon a routinely generated CV entanglement, namely, the two-mode squeezed vacuum (TMSV) state, the resulting Wigner function is positive-definite and as such, the TMSV state cannot violate any Bell inequality using CV quadrature measurements. We show here, however, that a Bell inequality for CV states in terms of entropies can be quantum mechanically violated by the TMSV state with two coarse-grained quadrature measurements per site within experimentally accessible parameter regime. The proposed CV entropic Bell inequality is advantageous for an experimental test, especially for a possible loophole-free test of nonlocality, as the quadrature measurements can be implemented with homodyne detections of nearly 100100%100 detection efficiency under current technology.