Operator Spaces, Linear Logic and the Heisenberg-Schr\"odinger Duality of Quantum Theory

TL;DR

This paper demonstrates that the category of operator spaces models intuitionistic and classical linear logic, aligning with quantum dualities and providing a framework for quantum information and higher-order quantum maps.

Contribution

It introduces a model of linear logic based on operator spaces that aligns with quantum dualities and supports advanced quantum information concepts.

Findings

OS is locally countably presentable.

OS models classical linear logic with duality compatible with quantum theory.

OS is suitable for studying quantum states, information, and higher-order maps.

Abstract

We show that the category OS of operator spaces, with complete contractions as morphisms, is locally countably presentable and a model of Intuitionistic Linear Logic in the sense of Lafont. We then describe a model of Classical Linear Logic, based on OS, whose duality is compatible with the Heisenberg-Schr\"odinger duality of quantum theory. We also show that OS provides a good setting for studying pure state and mixed state quantum information, the interaction between the two, and even higher-order quantum maps such as the quantum switch.