Lower and upper probabilities in the distributive lattice of subsystems

TL;DR

This paper explores how the structure of subsystems in finite quantum systems forms a distributive lattice and demonstrates that lower and upper probabilities, as understood in Dempster-Shafer theory, are relevant to quantum formalism.

Contribution

It establishes a connection between Dempster-Shafer theory and quantum subsystem lattices, showing the applicability of lower and upper probabilities in quantum contexts.

Findings

The subsystem lattice obeys a supermodularity inequality.

Lower probabilities in this context align with Dempster-Shafer theory.

Upper probabilities relate to the multivaluedness in quantum formalism.

Abstract

The set of subsystems of a finite quantum system (with variables in Z(n)) together with logical connectives, is a distributive lattice. With regard to this lattice, the (where P(m) is the projector to) obeys a supermodularity inequality, and it is interpreted as a lower probability in the sense of the Dempster-Shafer theory, and not as a Kolmogorov probability. It is shown that the basic concepts of the Dempster-Shafer theory (lower and upper probabilities and the Dempster multivaluedness) are pertinent to the quantum formalism of finite systems.