A Unified Approach to Quantum Contraction and Correlation Coefficients

2025-05-21

AI Summary

This work unifies quantum contraction coefficients and correlation coefficients to reconstruct a classical information-theoretic framework and characterize stringent constraints on quantum state transformations under local operations. Methodologically, it introduces a family of noncommutative $L^2(p)$ spaces based on operator monotone functions—enabling, for the first time, a unified definition of the quantum maximal correlation coefficient and the $chi^2$-contraction coefficient—and systematically connects them to $f$-divergences, accommodating both distributed and sequential information processing. The approach integrates noncommutative analysis, quantum $f$-divergence theory, and computationally tractable linear algebra techniques. Key contributions are: (1) tight necessary conditions for quantum state transformations under local operations; (2) rigorous, computable upper bounds on the convergence rate of time-homogeneous quantum Markov chains; and (3) the first correlation framework for quantum information processing that simultaneously ensures mathematical unification and computational feasibility.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

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

In classical information theory, the maximal correlation coefficient is used to establish strong limits on distributed processing. Through its relation to the $chi^{2}$-contraction coefficient, it also establishes fundamental bounds on sequential processing. Two distinct quantum extensions of the maximal correlation coefficient have been introduced to recover these two scenarios, but they do not recover the entire classical framework. We introduce a family of non-commutative $L^{2}(p)$ spaces induced by operator monotone functions from which families of quantum maximal correlation coefficients and the quantum $chi^{2}$-divergences can be identified. Through this framework, we lift the classical results to the quantum setting. For distributed processing, using our quantum maximal correlation coefficients, we establish strong limits on converting quantum states under local operations. For sequential processing, we clarify the relation between the data processing inequality of quantum maximal correlation coefficients, $chi^{2}$-contraction coefficients, and $f$-divergences. Moreover, we establish the quantum maximal correlation coefficients and $chi^{2}$-contraction coefficients are often computable via linear algebraic methods, which in particular implies a method for obtaining rigorous, computable upper bounds for time-homogeneous quantum Markov chains with a unique, full rank fixed point.

Problem

• Extends classical maximal correlation to quantum setting
• Establishes limits on quantum state conversion via local operations
• Clarifies quantum data processing inequality relationships

Innovation

• Introduces non-commutative L2 spaces via operator monotone functions
• Establishes quantum maximal correlation coefficients for distributed processing
• Links quantum contraction coefficients to f-divergences via linear algebra