On the lattice structure of probability spaces in quantum mechanics

TL;DR

This paper explores the lattice structure of quantum state spaces, unifying various quantum phenomena and extending separability criteria to infinite dimensions, with implications for generalized statistical theories.

Contribution

It introduces a lattice-theoretic framework for convex subsets of quantum states, unifying entanglement, Max-Ent principles, and extending separability criteria to infinite-dimensional and generalized models.

Findings

Lattice structure unifies quantum entanglement and Max-Ent principles.

New separability criterion expressed in lattice language.

Extension of convex polytope criteria to infinite-dimensional spaces.

Abstract

Let C be the set of all possible quantum states. We study the convex subsets of C with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets concerning the geometrical structure of the convex subsets. It is seen that the above mentioned framework is also capable of generalization to any statistical theory via the so-called convex operational models' approach. In particular, we show how to extend the geometrical structure underlying entanglement to any statistical model, an extension which may be useful for studying correlations in different generalizations of quantum mechanics.