A substructural logic for quantum measurements

TL;DR

This paper introduces a specialized substructural logic tailored for quantum measurements, incorporating unique connectives and semantics that reflect quantum orthogonality and projection, offering a novel logical framework for quantum theory.

Contribution

It develops a new substructural logic with restricted exchange and weakening, complete with semantics that model quantum orthogonality and projection operations.

Findings

Logic is sound for sequences of quantum measurements.

Semantic structures include orthogonality relations.

Provides a proof-theoretic framework for quantum measurement logic.

Abstract

This paper presents a substructural logic of sequents with very restricted exchange and weakening rules. It is sound with respect to sequences of measurements of a quantic system. A sound and complete semantics is provided. The semantic structures include a binary relation that expresses orthogonality between elements and enables the definition of an operation that generalizes the projection operation in Hilbert spaces. The language has a unitary connective, a sort of negation, and two dual binary connectives that are neither commutative nor associative, sorts of conjunction and disjunction. This provides a logic for quantum measurements whose proof theory is aesthetically pleasing.