The Quantum Logic of Direct-Sum Decompositions

TL;DR

This paper explores a dual perspective on quantum logic, focusing on direct-sum decompositions of vector spaces, which broadens the understanding of measurement beyond traditional subspace-based approaches, especially in finite-dimensional cases.

Contribution

It introduces the quantum logic of direct-sum decompositions, extending the traditional subspace logic to include measurement by self-adjoint operators in a dual framework.

Findings

Develops a logical framework for direct-sum decompositions

Analyzes the case of vector spaces over GF(2) for quantum models

Provides combinatorial analysis of decompositions in finite vector spaces

Abstract

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over 2 where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF(q) is analyzed with computations for the case of QM/Sets where q=2.