A Bell Inequality Analog in Quantum Measure Theory

Abstract

One obtains Bell's inequalities if one posits a hypothetical joint probability distribution, or {\it measure}, whose marginals yield the probabilities produced by the spin measurements in question. The existence of a joint measure is in turn equivalent to a certain causality condition known as ``screening off''. We show that if one assumes, more generally, a joint {\it quantal measure}, or ``decoherence functional'', one obtains instead an analogous inequality weaker by a factor of $\sqrt{2}$. The proof of this ``Tsirel'son inequality'' is geometrical and rests on the possibility of associating a Hilbert space to any strongly positive quantal measure. These results lead both to a {\it question}: ``Does a joint measure follow from some quantal analog of `screening off'?'', and to the {\it observation} that non-contextual hidden variables are viable in histories-based quantum mechanics, even if they are excluded classically.

38 pages, TeX. Several changes and added comments to bring out the meaning more clearly. Minor rewording and extra acknowledgements, now closer to published version

References in corpus (3)

• A Classical Sequential Growth Dynamics for Causal Sets
• Anomalies of weakened decoherence criteria for quantum histories
• The Random Walk in Generalized Quantum Theory

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