Quantum Reverse Shannon Theorem Revisited

2025-04-09

AI Summary

This work addresses the long-standing separation between classical and quantum reverse Shannon theorems in channel simulation, proposing the first unified quantum reverse Shannon theorem. Methodologically, it integrates entanglement-assisted communication, typical subspace analysis, Stinespring dilation, and quantum Markov chain modeling to extend the theorem to general correlated settings with arbitrary mixed-state reference systems, while simultaneously accommodating both feedback and non-feedback coding structures. Key contributions include: (i) the first complete unification across three orthogonal dimensions—classical/quantum, feedback/non-feedback, and pure/mixed states; (ii) a rigorous characterization of the optimal noise rate for simulating arbitrary quantum channels under arbitrary auxiliary correlations; and (iii) a definitive resolution of the open unification problem posed by Bennett et al., thereby establishing a universal foundational framework for quantum channel simulation theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.

Problem

• Unify classical and quantum channel simulation theorems
• Extend theorem to general mixed-state reference systems
• Unify feedback and non-feedback simulation approaches

Innovation

• Unifies classical and quantum channel simulations
• Extends theorem with mixed-state reference system
• Combines feedback and non-feedback theorems