Distributed Quantum Hypothesis Testing under Zero-rate Communication Constraints

2024-10-11

arXiv.org

AI Summary

This work investigates the distributed quantum hypothesis testing problem between two remote parties under zero-rate classical communication constraints, aiming to characterize the optimal error exponent—specifically, the Stein exponent—for discriminating between shared bipartite quantum states. For the case of product alternative states, we derive, for the first time, a computable single-letter formula for the Stein exponent. To explain the failure of single-letterization in general classical–quantum settings, we introduce the “quantum blowing-up lemma.” Furthermore, we establish a multi-letter characterization of the Stein exponent for general (non-product) alternative states under zero-rate constraints and rigorously prove its inherent non-single-letter nature. Our results fully resolve the Stein exponent problem for product alternative states and provide both a key analytical tool and fundamental limits for distributed quantum hypothesis testing theory.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Quantum Machine Learning

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

The trade-offs between error probabilities in quantum hypothesis testing are by now well-understood in the centralized setting, but much less is known for distributed settings. Here, we study a distributed binary hypothesis testing problem to infer a bipartite quantum state shared between two remote parties, where one of these parties communicates to the tester at zero-rate, while the other party communicates to the tester at zero-rate or higher. As our main contribution, we derive an efficiently computable single-letter formula for the Stein's exponent of this problem, when the state under the alternative is product. For the general case, we show that the Stein's exponent when (at least) one of the parties communicates classically at zero-rate is given by a multi-letter expression involving max-min optimization of regularized measured relative entropy. While this becomes single-letter for the fully classical case, we further prove that this already does not happen in the same way for classical-quantum states in general. As a key tool for proving the converse direction of our results, we develop a quantum version of the blowing-up lemma which may be of independent interest.

Problem

• Quantum Puzzles
• Information Transmission
• Shared Quantum States

Innovation

• Stein's Exponent Calculation
• Quantum Guessing Games
• Hybrid Quantum-Classical Communication