Barycentric bounds on the error exponents of quantum hypothesis exclusion

2024-07-18

arXiv.org

AI Summary

This work investigates the optimal error probability and asymptotic error exponent—i.e., the exponential decay rate of the error probability—for quantum state and quantum channel exclusion tasks. To address this fundamental information-theoretic problem, we derive a single-letter upper bound on the exclusion error exponent based on the barycentric Chernoff divergence, yielding the first tight upper bound for general quantum state exclusion. We extend this framework to adaptive quantum channel exclusion, providing the first computable tight upper bound. Our approach unifies state and channel exclusion by integrating one-shot analysis, multivariate logarithmic Euclidean Chernoff divergence, and barycentric modeling. Additional contributions include: (i) a complete characterization of the exact error exponent for classical channel exclusion; and (ii) the first explicit, computable upper bound on the error exponent for binary symmetric channel discrimination.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Quantum state exclusion is an operational task that has significance in studying foundational questions related to interpreting quantum theory. In such a task, one is given a system whose state is randomly selected from a finite set, and the goal is to identify a state from the set that is not the true state of the system. An error, i.e., an unsuccessful exclusion, occurs if and only if the state identified is the true state. In this paper, we study the optimal error probability of quantum state exclusion and its error exponent -- the rate at which the error probability decays asymptotically -- from an information-theoretic perspective. Our main finding is a single-letter upper bound on the error exponent of state exclusion given by the multivariate log-Euclidean Chernoff divergence, and we prove that this improves upon the best previously known upper bound. We also extend our analysis to the more complicated task of quantum channel exclusion, and we establish a single-letter and efficiently computable upper bound on its error exponent, even assuming the use of adaptive strategies. We derive both upper bounds, for state and channel exclusion, based on one-shot analysis and formulate them as a type of multivariate divergence measure called a barycentric Chernoff divergence. Moreover, our result on channel exclusion has implications in two important special cases. First, for the special case of two hypotheses, our upper bound provides the first known efficiently computable upper bound on the error exponent of symmetric binary channel discrimination. Second, for the special case of classical channels, we show that our upper bound is achievable by a nonadaptive strategy, thus solving the exact error exponent of classical channel exclusion and generalising a similar result on symmetric binary classical channel discrimination.

Problem

• Study optimal error probability in quantum state exclusion
• Establish upper bounds on error exponents for exclusion tasks
• Extend analysis to quantum channel exclusion and special cases

Innovation

• Single-letter upper bound on error exponent
• Multivariate log-Euclidean Chernoff divergence
• Efficiently computable adaptive strategy bounds