Positive Phase Space Transformation Incompatible with Classical Physics

Bell conjectured that a positive Wigner function does not allow violation of the inequalities imposed by local hidden variable theories. A requirement for this conjecture is “when phase space measurements are performed.” We introduce the theory-independent concept of “operationally local transformations” which refers to the change of the switch on a local measurement apparatus. We show that two separated parties, performing only phase space measurements on a composite quantum system with a positive Wigner function and performing only operationally local transformations that preserve this positivity, can nonetheless violate Bell’s inequality. Such operationally local transformations are realized using entangled ancillae.

Figure 1

Wigner functions $W_{ij}$ of the density matrix $\widehat{\rho }({\theta }_i,{\phi }_j)$, Eq. (7), for the transformations (${\theta }_i$, ${\phi }_j$) with $i$, $j=1$, 2 and $({\theta }_1,{\theta }_2,{\phi }_1,{\phi }_2)=(0,\pi /4,\pi /8,-\pi /8)$. The $W_{ij}$ are drawn for $\alpha =2$ as a function of the real parts of Alice’s and Bob’s hidden variables ${\beta }_1$ and ${\beta }_2$, respectively.