# Arbitrarily large violations of non-contextuality in single mode photon states with positive Wigner function. (arXiv:1807.02762v4 [quant-ph] UPDATED)

Banaszek, W\'odkiewicz and others

(\cite{Banaszek},\cite{Chen},\cite{Chen-Zhang}) made the surprising discovery

that Einstein-Bell locality inequalities can be violated by the two mode

squeezed vacuum by a factor $\sqrt{2}$, 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 \cite{Gleason} 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 $0.842

(\sqrt{2})^{N-1} $ for odd $N \geq 3$ . 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 $N$.