Arbitrarily large violations of non-contextuality in single mode photon states with positive Wigner function

Abstract

Banaszek, Wódkiewicz and others (Banaszek and Wdkiewicz 1998 Phys. Rev. A 58 4345; Chen et al 2002 Phys. Rev. Lett. 88 040406; Chen and Zhang 2002 Phys. Rev. A 65 044102) made the surprising discovery that Einstein–Bell locality inequalities can be violated by the two mode squeezed vacuum by a factor , in spite of the fact that the state has a positive Wigner function. I use here the more general Gleason–Kochen–Specker assumption of non-contextuality (Gleason 1957 J. Math. Mech. 6 885) to express classicality. I then derive non-contextuality Bell inequalities for correlations of N pseudo spins embedded in an infinite dimensional continuous variable Hilbert space, and show that their maximum possible quantum violation is by a factor 2(N−1)/2. I find quantum states for which this maximum violation is reached. I also show that the familiar displaced squeezed vacuum for a single optical mode, which has a positive Wigner function, can violate the inequality by a factor for odd . The arbitrarily large non-classicality means that realizations of the pseudo-spin measurements even in a single mode photon state might afford similar opportunities in quantum information tasks as entangled N qubit systems with large N.