IMALL with a Mixed-State Modality: A Logical Approach to Quantum Computation

2025-06-11

AI Summary

This paper addresses the lack of a logical foundation for mixed-state quantum computation by introducing the first intuitionistic multiplicative-additive linear logic (IMALL) system tailored to mixed states. Methodologically, it incorporates a modal operator B, interpreted via a categorical functor semantics that unifies pure and mixed states, supporting algebraic operations such as superposition and measurement. Crucially, quantum measurement is realized as a definable term within IMALL—eliminating reliance on explicit quantum configurations. The system is modeled over the category of C*-algebras and Hilbert spaces, establishing algebraic term formation and compositional reduction relations, and proving cut elimination. Theoretically, it fully expresses linear maps on ℂ² and successfully encodes prototypical quantum protocols—including quantum teleportation and the quantum switch—while preserving reduction soundness under denotational semantics.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Logic Programming
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements

Abstract

We introduce a proof language for Intuitionistic Multiplicative Additive Linear Logic (IMALL), extended with a modality B to capture mixed-state quantum computation. The language supports algebraic constructs such as linear combinations, and embeds pure quantum computations within a mixed-state framework via B, interpreted categorically as a functor from a category of Hilbert Spaces to a category of finite-dimensional C*-algebras. Measurement arises as a definable term, not as a constant, and the system avoids the use of quantum configurations, which are part of the theory of the quantum lambda calculus. Cut-elimination is defined via a composite reduction relation, and shown to be sound with respect to the denotational interpretation. We prove n that any linear map on C 2 can be represented within the system, and illustrate this expressiveness with examples such as quantum teleportation and the quantum switch.

Problem

• Extends IMALL with modality B for mixed-state quantum computation
• Embeds pure quantum computations in mixed-state framework via B
• Represents any linear map on C^2, enabling quantum protocols

Innovation

• Extended IMALL with mixed-state modality B
• Algebraic constructs support linear combinations
• Measurement as definable term, no configurations