Technology Category
Machine Learning: Quantum Machine LearningKnowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
On the undecidability of quantum channel capacities
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
quantum channel capacities
uncomputability
QMA-hard
zero-error capacity
computability
Innovation
Methods, ideas, or system contributions that make the work stand out.
quantum channel capacity
QMA-hardness
uncomputability
zero-error capacity
entanglement-assisted communication
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