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

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.

Figure 1

The plot of ${\mathcal{D}}_{\mathrm{QM}}$ [see Eq. (27)] as a function of angle $\delta$ and the squeezing parameter $r$ for different values of the measurement resolutions $\Delta$. We also show the plane of ${\mathcal{D}}_{\mathrm{QM}}=0$, above which is the parameter regime of violating the inequality Eq. (10). An apparent trend is that larger $\Delta$ requires smaller $r$ to violate the entropic Bell inequality.

Figure 2

The plot of ${\mathcal{D}}_{\mathrm{QM}}$ at $\delta =0$ as a function of the squeezing parameter $r$ and the measurement resolution $\Delta$. The parameter regime violating the inequality Eq. (10) is above the plane of ${\mathcal{D}}_{\mathrm{QM}}(r,\Delta ,\delta =0)=0$.