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

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 $2^{(N-1)\mathrm{/}2}$. 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} $forodd$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 $N$.