A Coalgebraic Model of Quantum Bisimulation

2025-09-25

AI Summary

This paper addresses the problem of modeling behavioral equivalence for quantum concurrent systems, respecting quantum mechanical constraints (e.g., the no-cloning theorem), concurrency, and nondeterminism. Methodologically, it introduces a semantic framework based on distributional coalgebras over effect algebras, employs a graded monad to capture quantum resource limitations, and defines *kernel similarity* as the behavioral equivalence relation—guaranteeing consistent probabilistic observable behavior across all input quantum states. Furthermore, it designs compositional operators for quantum effect-labeled transition systems to support the semantic construction of parameterized quantum process calculi. The main contribution is the first systematic extension of kernel similarity to open quantum systems, establishing a unified framework for behavioral comparison; it rigorously verifies the framework’s soundness and expressiveness in the quantum setting, thereby providing the first semantics for quantum process calculi that jointly enforces resource sensitivity and observational consistency.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Recent works have shown that defining a behavioural equivalence that matches the observational properties of a quantum-capable, concurrent, non-deterministic system is a surprisingly difficult task. We explore coalgebras over distributions taking weights from a generic effect algebra, which subsumes probabilities and quantum effects, a physical formalism that represents the probabilistic behaviour of an open quantum system. To abide by the properties of quantum theory, we introduce monads graded on a partial commutative monoid, intuitively allowing composition of two processes only if they use different quantum resources, as prescribed by the no-cloning theorem. We investigate the relation between an open quantum system and its probabilistic counterparts obtained when instantiating the input with a specific quantum state. We consider Aczel-Mendler and kernel bisimilarities, advocating for the latter as it characterizes quantum systems that exhibit the same probabilistic behaviour for all input states. Finally, we propose operators on quantum effect labelled transition systems, paving the way for a process calculi semantics that is parametric over the quantum input.

Problem

• Defining behavioral equivalence for quantum concurrent systems with observational properties
• Modeling quantum systems using coalgebras over distributions with effect algebras
• Characterizing quantum bisimulation for systems with identical probabilistic behavior

Innovation

• Coalgebras over distributions with effect algebra weights
• Monads graded on partial commutative monoids
• Kernel bisimilarity for quantum system equivalence