Exponential Modalities and Complementarity (extended abstract)

TL;DR

This paper explores how exponential modalities in linear logic can model quantum complementarity, linking logical structures to quantum measurement and categorical quantum mechanics.

Contribution

It demonstrates how exponential modalities in linear logic give rise to quantum complementarity through categorical semantics and measurement processes.

Findings

Exponential modalities can model quantum complementarity.

Measurement on free exponential modalities produces complementary systems.

Categorical semantics relate exponential modalities to Frobenius algebras.

Abstract

The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics. The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.