A Unified Approach to Quantum Contraction and Correlation Coefficients

TL;DR

This paper develops a quantum extension of classical correlation and contraction measures, introducing computable quantum coefficients and establishing a fundamental relation between them, with applications to quantum channel mixing times.

Contribution

It constructs a unified quantum framework for maximal correlation and contraction coefficients, extending classical properties and providing efficient algorithms for quantum information processing.

Findings

Quantum maximal correlation coefficients impose limits on quantum state conversions.

A family of quantum contraction coefficients are efficiently computable.

A quantum analogue of Raginsky's classical correspondence is established.

Abstract

The maximal correlation coefficient measures the linear correlation in a bipartite distribution and contraction coefficients measure how much information is lost under a noisy channel. Remarkably, Raginsky established a close relation between these two concepts by showing that the $\chi^2$ contraction coefficient equals the maximal correlation coefficient of the joint input/output distribution of the channel. In quantum theory, several generalizations of these concepts have been proposed, but none recover all the classical properties. Here we construct a framework in which the classical theory extends to the quantum setting. We introduce families of quantum maximal correlation coefficients and show that many impose limits on converting quantum states under local operations. We establish a family of quantum contraction coefficients are efficiently computable, yielding a generic efficient algorithm for mixing times of quantum channels with a full rank fixed point. Furthermore, we establish a quantum analogue of Raginsky's classical correspondence that relates these two families of quantities. To do this, we develop the operator-theoretic approach to Petz's family of non-commutative $L^{2}(p)$ spaces that extend the data processing inequality for variance to quantum theory.