  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.
