Non-Deterministic Semantics for Quantum States

Abstract

In this work, we discuss the failure of the principle of truth functionality in the quantum formalism. By exploiting this failure, we import the formalism of N-matrix theory and non-deterministic semantics to the foundations of quantum mechanics. This is done by describing quantum states as particular valuations associated with infinite non-deterministic truth tables. This allows us to introduce a natural interpretation of quantum states in terms of a non-deterministic semantics. We also provide a similar construction for arbitrary probabilistic theories based in orthomodular lattices, allowing to study post-quantum models using logical techniques.

Generalized Probabilities in Statistical Theories

“…Thus, for example, if we assign the truth value 1 to P 0,0,0,1 (i.e., ν(P 0,0,0,1 ) = 1), then, all other members of that context must have the truth value 0 assigned (ν(P 0,0,1,0 ) = ν(P 1,1,0,0 ) = ν(P 1,−1,0,0 ) = 0). Equation (20) implies that the valuations must satisfy ∑ i ν(P i ) = 1 on each line (this is known as the FUNC condition in the literature; see the discussion and references in [98]). We must assume that the valuations preserve their values from context to context (if we assign a certain truth value to a projection in a given context, we must use that same value when it appears in a different context).…”

“…This example illustrates clearly how the Boolean algebras of events associated to quantum systems are intertwined and how this complex structure gives place to interpretational issues. As is well known, the Kochen-Specker theorem is a cornerstone in the discussions about the foundations of quantum mechanics (see, for example [98] and the references therein).…”

On the Connection Between Quantum Probability and Geometry

“…The intertwining of the Boolean subalgebras of P(H) is behind the Kochen-Specker contextuality (see [46][47][48]). According to the Kochen-Specker theorem, there is no algebra homomorphism between P(H) and the two valued Boolean algebra {0, 1} (see, for example, [49] for details).…”

On the assumptions underlying KS-like contradictions

“…In words: each hidden state assigns an homomorphism v λ : L(H) −→ {0, 1}. This means that v λ assigns truth values (0 or 1) to each proposition in L(H) in a functional way (see [26] for the technical meaning of "functional"). Let us illustrate how this last condition (and 6) work in a given measurement context.…”