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arXiv:1006.4012 · Physical Review A

High-dimensional Bell test for a continuous-variable state in phase space and its robustness to detection inefficiency

Seung-Woo Lee, Hyunseok Jeong

Published Feb 10, 2011• 4 citations
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Abstract

We propose a scheme for testing high-dimensional Bell inequalities in phase space. High-dimensional Bell inequalities can be recast into the forms of a phase-space version using quasiprobability functions with the complex-valued order parameter. We investigate their violations for two-mode squeezed states while increasing the dimension of measurement outcomes and finally show the robustness of high-dimensional tests to detection inefficiency.

DOI
10.1103/physreva.83.022103

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We propose a scheme for testing high-dimensional Bell inequalities in phase space. High-dimensional Bell inequalities can be recast into the forms of a phase-space version using quasiprobability functions with the complex-valued order parameter. We investigate their violations for two-mode squeezed states while increasing the dimension of measurement outcomes and finally show the robustness of high-dimensional tests to detection inefficiency.
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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.
We investigate generalized Bell tests in phase space for continuous variables that can be used to certify quantum states and witness the dimension of a given state. Our so-called s-parametrized observables in the Bell tests correspond to measuring excitation (photon) -number distribution in conjunction with phase-space displacements. Specifically, we study three practically important classes of states broadly, i.e. two-mode squeezed states, entangled coherent states and general entangled state with bounded photon numbers. We obtain the degree of optimal violation of Bell tests for each class, which enables us to certify a quantum state yielding a maximal violation. Furthermore, our analysis also makes it possible to witness the dimension of a quantum state, i.e. the photon number, by obtaining the bound of Bell violation for a different total photon number.
We propose a Bell inequality for high-dimensional bipartite systems obtained by binning local measurement outcomes and show that it is tight. We find a binning method for even ddd-dimensional measurement outcomes for which this Bell inequality is maximally violated by maximally entangled states. Furthermore, we demonstrate that the Bell inequality is applicable to continuous variable systems and yields strong violations for two-mode squeezed states.
We derive a Bell inequality based on a generalized quasiprobability function, which is parametrized by one nonpositive real value. Two types of known Bell inequalities formulated in terms of the Wigner and the QQQ functions are included as limiting cases. We investigate violations of our Bell inequalities for single-photon entangled states and two-mode squeezed vacuum states when varying the detector efficiency. We show that the Bell inequality for the QQQ function allows the lowest detection efficiency for violations of local realism.