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arXiv:1409.0437 · Physical Review A

Violations of entropic Bell inequalities with coarse-grained quadrature measurements for continuous-variable states

Zeng‐Bing Chen, Yao Fu, Yu-Kang Zhao

Published Aug 26, 2014• 5 citations
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Abstract

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 $100%$ detection efficiency under current technology.

DOI
10.1103/physreva.90.022124

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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.
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.
When separated measurements on entangled quantum systems are performed, the theory predicts correlations that cannot be explained by any classical mechanism: communication is excluded because the signal should travel faster than light; preestablished agreement is excluded because Bell inequalities are violated. All optical demonstrations of such violations have involved discrete degrees of freedom and are plagued by the detection-efficiency loophole. A promising alternative is to use continuous variables combined with highly efficient homodyne measurements. However, all the schemes proposed so far use states or measurements that are extremely difficult to achieve, or they produce very weak violations. We present a simple method to generate large violations for feasible states using both photon counting and homodyne detections. The present scheme can also be used to obtain nonlocality from easy-to-prepare Gaussian states (e.g., two-mode squeezed state).
When separated measurements on entangled quantum systems are performed, the theory predicts correlations that cannot be explained by any classical mechanism: communication is excluded because the signal should travel faster than light; pre-established agreement is excluded because Bell inequalities are violated. All optical demonstrations of such violations have involved discrete degrees of freedom and are plagued by the detection-efficiency loophole. A promising alternative is to use continuous variables combined with highly efficient homodyne measurements. However, all schemes proposed so far use states or measurements that are extremely difficult to achieve, or produce very weak violations. Here we show that large violations for feasible states can be achieved if both photon counting and homodyne detections are used. Our scheme may lead to the first violation of Bell inequalities using continuous-variable measurements and pave the way for a loophole-free Bell test.
While commonly used entanglement criteria for continuous variable systems are based on quadrature measurements, here we study entanglement detection from measurements of the Wigner function. These are routinely performed in platforms such as trapped ions and circuit QED, where homodyne measurements are difficult to be implemented. We provide complementary criteria which we show to be tight for a variety of experimentally relevant Gaussian and non-Gaussian states. Our results show novel approaches to detect entanglement in continuous variable systems and shed light on interesting connections between known criteria and the Wigner function.
We derive a general condition for nonlocality using the Banaszek-Wódkiewicz phase-space Wigner representation of Bell's function in continuous-variable Gaussian states, showing rigorously that while entanglement is necessary, it is not sufficient for nonlocality, whereas nonlocality always implies entanglement. Furthermore, to demonstrate the applicability of the framework, we analyze the two-mode squeezed output generated by a double-cavity optomechanical system coupled to a common mechanical resonator-an experimentally accessible hybrid electro/optomechanical interface. We show that maximal entanglement does not necessarily lead to Bell violation, whereas nonlocality may arise in states with lower entanglement, highlighting the crucial role of mixedness. The developed formalism is universally applicable to all bipartite Gaussian states and provides a general tool for investigating foundational and operational aspects of nonlocality in continuous-variable quantum platforms.
We have theoretically investigated the possibility of using any of several continuous-variable Bell-type inequalities---for which the dichotomic measurements are achieved with coarse-grained quadrature (homodyne) measurements---in a multiparty configuration where each participant is given a section, in the frequency domain, of the output of an optical parametric oscillator which has been synchronously pumped with a frequency comb. Such light sources are undergoing intense study due to their novel properties, including the potential for production of light entangled in many hundreds of physical modes---a critical component for many proposals in optical or hybrid-optical quantum computation proposals. The situation we study notably uses only highly efficient optical homodyne detection, meaning that in such systems the fair-sampling loophole would be relatively easy to avoid.