Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach

TL;DR

This paper develops a categorical model of MALL using von Neumann algebras, linking quantum operations with logical proofs through a duality inspired by quantum theory.

Contribution

It introduces a novel von Neumann algebraic framework for modeling MALL, connecting logical polarity with quantum channels in a categorical setting.

Findings

Proofs of positive polarity correspond to CPTP maps.

Proofs of negative polarity correspond to CPU maps.

The model employs noncommutative geometry and finite-dimensional von Neumann algebras.

Abstract

We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schr\"odinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schr\"odinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.