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

Bert Lindenhovius; Vladimir Zamdzhiev

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arXiv · 2505.06069

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

Bert Lindenhovius·Vladimir Zamdzhiev

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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.

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Bert Lindenhovius, Vladimir Zamdzhiev. 2025-05-09. Operator Spaces, Linear Logic and the Heisenberg-Schr\"odinger Duality of Quantum Theory. https://arxiv.org/abs/2505.06069

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