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Operator Spaces, Linear Logic and the Heisenberg-Schrödinger Duality of Quantum Theory

Bert Lindenhovius, Vladimir Zamdzhiev

TL;DR

This paper forges a bridge between quantum dualities and linear logic by modeling quantum-theoretic dualities in the category of operator spaces $oldsymbol{OS}$. It proves that $oldsymbol{OS}$ is locally countably presentable, symmetric monoidal closed, and supports a Lafont-style exponential, yielding a model of ILL; it also yields a Chu construction-based model of full CLL whose duality mirrors the Heisenberg-Schrödinger duality. The framework elegantly represents both pure and mixed quantum information, including higher-order maps such as the quantum switch, and clarifies how different quantum compositions correspond to polarities in LL. By connecting noncommutative geometry with LL semantics, the work sets a foundation for semantic and logical analysis of quantum systems and their higher-order behaviors. The results motivate future work on restricted map classes (CPTP/NCPU or CPTNI/NCPSU), Haagerup-tensor phenomena, BV-logic connections, and language-theoretic implications for quantum programming.

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-Schrödinger 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.

Operator Spaces, Linear Logic and the Heisenberg-Schrödinger Duality of Quantum Theory

TL;DR

This paper forges a bridge between quantum dualities and linear logic by modeling quantum-theoretic dualities in the category of operator spaces . It proves that is locally countably presentable, symmetric monoidal closed, and supports a Lafont-style exponential, yielding a model of ILL; it also yields a Chu construction-based model of full CLL whose duality mirrors the Heisenberg-Schrödinger duality. The framework elegantly represents both pure and mixed quantum information, including higher-order maps such as the quantum switch, and clarifies how different quantum compositions correspond to polarities in LL. By connecting noncommutative geometry with LL semantics, the work sets a foundation for semantic and logical analysis of quantum systems and their higher-order behaviors. The results motivate future work on restricted map classes (CPTP/NCPU or CPTNI/NCPSU), Haagerup-tensor phenomena, BV-logic connections, and language-theoretic implications for quantum programming.

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-Schrödinger 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.
Paper Structure (19 sections, 29 theorems, 67 equations, 2 tables)

This paper contains 19 sections, 29 theorems, 67 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Categorical Preliminaries
  3. The Category of Operator Spaces
  4. Preliminaries on Operator Spaces
  5. Preliminaries on the Heisenberg-Schrödinger Duality
  6. Limits and Colimits in $\mathbf{OS}$
  7. Strong generators in $\mathbf{OS}$
  8. Local Presentability of $\mathbf{OS}$
  9. Symmetric Monoidal Closure of $\mathbf{OS}$
  10. Linear Logic Exponentials
  11. Pure and Mixed Quantum Information in $\mathbf{OS}$
  12. Morphisms for Mixed State Quantum Computation
  13. Interaction between Pure and Mixed Quantum Primitives
  14. Higher-order Maps with Superposition
  15. Linear Logic and the Heisenberg-Schrödinger Duality
  16. ...and 4 more sections

Key Result

Proposition 3.13

Let $H_1$ and $H_2$ be Hilbert spaces. The transpose operation $(-)^t$ gives a bijective correspondence between bounded linear maps $\psi \colon T(H_1) \to T(H_2)$ and normal linear maps $\psi^t \colon B(H_2) \to B(H_1)$. Furthermore:

Theorems & Definitions (71)

  • Definition 2.1: $\alpha$-directed Poset
  • Definition 2.2: $\alpha$-directed Diagram
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7: borceux2
  • Definition 3.1: Matrix Space
  • Definition 3.2
  • Definition 3.3
  • ...and 61 more