Independence and totalness of subspaces in phase space methods

TL;DR

This paper explores the concepts of independence and totalness of subspaces in phase space methods for quantum systems, revealing various levels of independence due to non-distributivity and introducing new formalizations and applications.

Contribution

It introduces a detailed framework for understanding subspace independence and totalness in quantum phase space, including duality and new formal approaches like Rota's formalism.

Findings

Multiple levels of independence identified in non-distributive lattice of subspaces

Duality between independence and totalness involving orthocomplementation

Application of concepts to analyze the pentagram in contextuality studies

Abstract

The concepts of independence and totalness of subspaces are introduced in the context of quasi-probability distributions in phase space, for quantum systems with finite-dimensional Hilbert space. It is shown that due to the non-distributivity of the lattice of subspaces, there are various levels of independence, from pairwise independence up to (full) independence. Pairwise totalness, totalness and other intermediate concepts are also introduced, which roughly express that the subspaces overlap strongly among themselves, and they cover the full Hilbert space. A duality between independence and totalness, that involves orthocomplementation (logical NOT operation), is discussed. Another approach to independence is also studied, using Rota's formalism on independent partitions of the Hilbert space. This is used to define informational independence, which is proved to be equivalent to independence. As an application, the pentagram (used in discussions on contextuality) is analyzed using these concepts.