On the undecidability of quantum channel capacities

2026-01-30

AI Summary

This study investigates the computability and computational complexity of quantum channel capacities. By leveraging tools from quantum complexity theory and undecidability analysis, it rigorously establishes for the first time that computing the quantum capacity of a general quantum channel is QMA-hard. Furthermore, it demonstrates that the zero-error one-shot classical capacity assisted by maximal entanglement is uncomputable. These results resolve a fundamental open question in quantum information theory concerning the computability of channel capacities and provide crucial theoretical underpinnings for quantum communication.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the maximal-entanglement-assisted zero-error one-shot classical capacity is uncomputable.

Problem

• quantum channel capacities
• uncomputability
• QMA-hard
• zero-error capacity
• computability

Innovation

• quantum channel capacity
• QMA-hardness
• uncomputability
• zero-error capacity
• entanglement-assisted communication