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Banaszek, W\'odkiewicz and others(\cite{Banaszek},\cite{Chen},\cite{Chen-Zhang}) made the surprising discoverythat Einstein-Bell locality inequalities can be violated by the two modesqueezed vacuum by a factor $\sqrt{2}$, in spite of the fact that the state hasa positive Wigner function. I use here the more general Gleason-Kochen-Speckerassumption of non-contextuality \cite{Gleason} to express classicality. I thenderive non-contextuality Bell inequalities for correlations of $N$ pseudo spinsembedded in an infinite dimensional continuous variable Hilbert space, and showthat their maximum possible quantum violation is by a factor $2^{(N-1)/2}$. Ifind quantum states for which this maximum violation is reached. I also showthat the familiar displaced squeezed vacuum for a single optical mode, whichhas 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-classicalitymeans that realizations of the pseudo-spin measurements even in a single modephoton state might afford similar opportunities in quantum information tasks asentangled $N$ qubit systems with large $N$.

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