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

TL;DR

This paper demonstrates that continuous-variable entangled states with positive Wigner functions can violate entropic Bell inequalities using coarse-grained quadrature measurements, challenging the traditional belief about nonlocality detection.

Contribution

It introduces an entropic Bell inequality for CV states and shows its violation with TMSV states using feasible homodyne detection, enabling loophole-free nonlocality tests.

Findings

TMSV states violate the entropic Bell inequality with coarse-grained measurements.

Violation occurs within experimentally accessible parameters.

Homodyne detection efficiency is nearly 100%, suitable for loophole-free tests.

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.