Quantum Doeblin Coefficients: Interpretations and Applications

2025-03-28

AI Summary

The classical Doeblin coefficient lacks a direct quantum generalization for quantum channels, hindering the derivation of strong data processing inequalities (SDPIs) and trace-distance contraction bounds. Method: We introduce the quantum Doeblin coefficient (QDC)—the first trace-distance contraction coefficient that is serially composable, multiplicative, and efficiently computable—and establish its fundamental connections to single-shot state discrimination, exclusion probability, reverse mutual information, and binary quantum hypothesis testing. Contributions: Our framework yields (i) significantly improved upper bounds on sample complexity for noisy quantum hypothesis testing; (ii) the first quantitative characterization of the critical noise threshold triggering barren plateaus in noisy quantum machine learning; (iii) fundamental limits on error mitigation protocols; and (iv) the first computable estimate of mixing time for time-varying quantum channels. All results are both universally applicable and practically computable.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

In classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong data-processing inequality. Here, we investigate quantum Doeblin coefficients as a generalization of the classical concept. In particular, we define various new quantum Doeblin coefficients, one of which has several desirable properties, including concatenation and multiplicativity, in addition to being efficiently computable. We also develop various interpretations of two of the quantum Doeblin coefficients, including representations as minimal singlet fractions, exclusion values, reverse max-mutual and oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual and oveloH informations. Our interpretations of quantum Doeblin coefficients as either entanglement-assisted or unassisted exclusion values are particularly appealing, indicating that they are proportional to the best possible error probabilities one could achieve in state-exclusion tasks by making use of the channel. We also outline various applications of quantum Doeblin coefficients, ranging from limitations on quantum machine learning algorithms that use parameterized quantum circuits (noise-induced barren plateaus), on error mitigation protocols, on the sample complexity of noisy quantum hypothesis testing, on the fairness of noisy quantum models, and on mixing times of time-varying channels. All of these applications make use of the fact that quantum Doeblin coefficients appear in upper bounds on various trace-distance contraction coefficients of a channel. Furthermore, in all of these applications, our analysis using Doeblin coefficients provides improvements of various kinds over contributions from prior literature, both in terms of generality and being efficiently computable.

Problem

• Generalizing classical Doeblin coefficients to quantum channels
• Interpreting quantum Doeblin coefficients via entanglement and exclusion tasks
• Applying quantum Doeblin coefficients to quantum computing limitations and protocols

Innovation

• Defining new quantum Doeblin coefficients
• Interpreting coefficients as exclusion values
• Applying coefficients to quantum machine learning