Computational Relative Entropy

2025-09-24

AI Summary

Classical quantum information theory lacks tools to quantify information under computational constraints, rendering standard measures like quantum relative entropy inadequate for resource-bounded settings. Method: We introduce *computational relative entropy*—a rigorously defined measure based on polynomial-size quantum circuits and sample complexity—unifying hypothesis testing, computational indistinguishability, and operational semantics. We further propose *computational smoothness* as a new structural concept. Contributions: (i) First proofs of computational analogues of Stein’s lemma and Pinsker’s inequality; (ii) A fundamental separation between computationally bounded and unbounded information quantities; (iii) A cryptographic separation between quantum and classical information processing; (iv) A *computational entropy* characterizing optimal compression rates under bounded computation; (v) A computational theory of entanglement, including a computationally constrained Rains bound. Collectively, these results establish the first systematic framework for computational quantum information theory.

Technology Category

• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Economic aspects and information design of Web, GenAI, and cloud computing

Abstract

Our capacity to process information depends on the computational power at our disposal. Information theory captures our ability to distinguish states or communicate messages when it is unconstrained with unrivaled elegance. For computationally bounded observers the situation is quite different. They can, for example, be fooled to believe that distributions are more random than they actually are. In our work, we go beyond the prevailing single-shot approach and take a new direction in computational quantum information theory that captures the essence of complexity-constrained information theory while retaining the look and feel of the unbounded asymptotic theory. As our foundational quantity, we define the computational relative entropy as the optimal error exponent in asymmetric hypothesis testing when restricted to polynomially many copies and quantum gates, defined in a mathematically rigorous way. Building on this foundation, we prove a computational analogue of Stein's lemma, establish computational versions of fundamental inequalities like Pinsker's bound, and demonstrate a computational smoothing property showing that computationally indistinguishable states yield equivalent information measures. We derive a computational entropy that operationally characterizes optimal compression rates for quantum states under computational limitations and show that our quantities apply to computational entanglement theory, proving a computational version of the Rains bound. Our framework reveals striking separations between computational and unbounded information measures, including quantum-classical gaps that arise from cryptographic assumptions, demonstrating that computational constraints fundamentally alter the information-theoretic landscape and open new research directions at the intersection of quantum information, complexity theory, and cryptography.

Problem

• Develops computational relative entropy for complexity-constrained quantum hypothesis testing
• Establishes computational versions of fundamental information theory inequalities and lemmas
• Reveals separations between computational and unbounded quantum information measures

Innovation

• Computational relative entropy for hypothesis testing
• Proving computational analogues of classical inequalities
• Framework revealing computational-unbounded information separations