A Bell inequality analog in quantum measure theory

Abstract

One obtains Bell's inequalities if one posits a hypothetical joint probability distribution, or 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 quantal measure, or "decoherence functional", one obtains instead an analogous inequality weaker by a factor of √ 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 question: "Does a joint measure follow from some quantal analog of 'screening off' ?", and to the observation that non-contextual hidden variables are viable in histories-based quantum mechanics, even if they are excluded classically.

Popescu–Rohrlich boxes in quantum measure theory

“…In section 3 we show that there is a set of inequalities on the correlations that is implied by the existence of a strongly positive joint decoherence functional or quantal measure on the (fictitious) joint sample space of all possible outcomes of both of Alya's and both of Bai's experiments. This tightens the bounds derived in reference [2].…”

“…Each PR box maximally violates one of the 8 CHSH inequalities. In [2] it was shown that the PR box admits a joint quantal measure, indeed a decoherence functional that does the job for one of the PR boxes is explicitly given in that paper. By doing the appropriate bit-flip on that decoherence functional (either flip the outcome of the a ′ measurement or swap b and b ′ : both produce the same result), we obtain the decoherence functional for the PR box (4.1): The remaining elements which are not equal to one of the above by Hermiticity are zero.…”

“…There are generalisations of the standard theory which yet remain at level 2. Even more surprisingly, one of these generalisations has been shown [2] to include the device that has come to be known as the "Popescu Rohrlich (PR) box" [3,4].…”

Hilbert spaces from path integrals

“…Second, strong positivity is preserved under composition of subsystems, whereas the obvious "product measure" of two weakly positive quantal measures is not in general positive at all. And third -at a technical level -the histories hilbert space to which strong positivity leads has already proven to be useful in certain applications [22,23], while there are also indications that the map taking events α ∈ A to vectors in H could be of aid in the effort to extend the decoherence functional from A to a larger fragment of the σ-algebra it generates.…”

How to Measure the Quantum Measure

“…In this paper we will work in the framework of Quantum Measure Theory [24,17,8,11,25], which generalizes the mathematical concept of a measure-space so as to allow for quantal interference. When refashioned in this language, Quantum Mechanics appears as a generalized probability theory of a type inspired by the path integral.…”