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 $N$.

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