Negation of the Tsirelson’s conjecture and genericity in quantum mechanics

Abstract

We show that the negation of the Tsirelson’s conjecture can be understood as, and in fact follows from, the ZFC - genericity of infinite sequences of the QM outcomes. We extend the result of Landsman that there holds the equivalence of the statistics of the entire sequences with the statistic of the single QM measurements in such sequences, over stratified infinite tensor products of the finite dimensional spaces. The stratification is based on uncomputable Turing classes and also on the Solovay genericity and the equivalence is now corrected by the appearance of certain self-adjoint operator. From studies on algorithmic randomness follows that these two properties are in a sense orthogonal one to the other for random sequences. We show two models of ZFC which indeed separate them. This is based on the classic results for Cohen and random forcings in set theory and also on Takeuti’s early results on Boolean-valued models and quantum mechanics. This separation is the tool for showing that not all correlations between commuting operators on infinite dimensional Hilbert spaces can be reproduced on a tensor product of two spaces with the corresponding self-adjoint operators. We discuss the findings also from the perspective of eventual practical applications. The supplementary subsections contain discussion of QM with the set theory component along with Solovay randomness of the outcomes.