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A Unified Approach to Quantum Contraction and Correlation Coefficients

Ian George, Marco Tomamichel

TL;DR

The paper advances a unified quantum framework that extends the classical Hirschfeld–Gebelein–Renyi maximal correlation μ(X:Y) and χ²-contraction to finite-dimensional quantum systems by embedding expectations and variances in non-commutative L²_f(σ) spaces induced by operator-monotone functions f. It introduces families of quantum maximal correlation coefficients μ_f and μ_f^{Lin}, relates them to χ²_f-divergences through operator-norm representations, and proves data-processing, isometric invariance, and (for GM) tensorization properties, enabling strong monotones for state conversions under local processing. It further shows how a CPTP map’s contraction coefficients η_{χ²_f}(𝓔,σ) quantify mixing rates of quantum Markov dynamics, with an exact Heisenberg/Schrödinger time-reversal duality, and provides efficient computability results via map-norm characterizations and eigenvalue problems. The work also establishes saturation conditions for data-processing inequalities, recoverability connections to Petz-type maps, and asymptotic data-distillation criteria, thereby generalizing classical results to the quantum regime. Practically, these tools yield efficient methods to bound mixing times and study asymptotic common data in quantum processes, with broad implications for quantum information processing and Markov dynamics.

Abstract

In classical information theory, the maximal correlation and $χ^{2}$-contraction coefficient establish limits on distributed and sequential processing. Two distinct quantum maximal correlation coefficients have been proposed, but they do not extend all the classical results. Building on work of Petz, we use the family of non-commutative $L^{2}(p)$ spaces that extend the data processing inequality for variance to quantum theory to extend the classical results to quantum theory. We introduce families of quantum maximal correlation coefficients and identify quantum $χ^{2}$-divergences as non-commutative generalizations of the variance of the likelihood ratio. We establish a family of maximal correlation coefficients that must all be ordered on a single copy level for an arbitrary number of copies of one state to be able to be converted to a single copy of another target state under local operations. We prove the equivalent characterizations of perfect classical correlation extraction via local operations in quantum theory. We clarify the relationship between maximal correlation and $χ^{2}$-contraction coefficients by proving they are the same operator norms evaluated on distinct maps. Then we establish new equivalent conditions to the saturation of the data processing inequality for $χ^{2}$-divergences. This implies previous saturation results for the $χ^{2}$ and sandwiched Rényi divergences. Finally, we establish the quantum maximal correlation coefficients and $χ^{2}$-contraction coefficients are often efficiently computable. This results in a generic method for efficiently computing mixing times of time-homogeneous quantum Markov chains with a unique full rank fixed point.

A Unified Approach to Quantum Contraction and Correlation Coefficients

TL;DR

The paper advances a unified quantum framework that extends the classical Hirschfeld–Gebelein–Renyi maximal correlation μ(X:Y) and χ²-contraction to finite-dimensional quantum systems by embedding expectations and variances in non-commutative L²_f(σ) spaces induced by operator-monotone functions f. It introduces families of quantum maximal correlation coefficients μ_f and μ_f^{Lin}, relates them to χ²_f-divergences through operator-norm representations, and proves data-processing, isometric invariance, and (for GM) tensorization properties, enabling strong monotones for state conversions under local processing. It further shows how a CPTP map’s contraction coefficients η_{χ²_f}(𝓔,σ) quantify mixing rates of quantum Markov dynamics, with an exact Heisenberg/Schrödinger time-reversal duality, and provides efficient computability results via map-norm characterizations and eigenvalue problems. The work also establishes saturation conditions for data-processing inequalities, recoverability connections to Petz-type maps, and asymptotic data-distillation criteria, thereby generalizing classical results to the quantum regime. Practically, these tools yield efficient methods to bound mixing times and study asymptotic common data in quantum processes, with broad implications for quantum information processing and Markov dynamics.

Abstract

In classical information theory, the maximal correlation and -contraction coefficient establish limits on distributed and sequential processing. Two distinct quantum maximal correlation coefficients have been proposed, but they do not extend all the classical results. Building on work of Petz, we use the family of non-commutative spaces that extend the data processing inequality for variance to quantum theory to extend the classical results to quantum theory. We introduce families of quantum maximal correlation coefficients and identify quantum -divergences as non-commutative generalizations of the variance of the likelihood ratio. We establish a family of maximal correlation coefficients that must all be ordered on a single copy level for an arbitrary number of copies of one state to be able to be converted to a single copy of another target state under local operations. We prove the equivalent characterizations of perfect classical correlation extraction via local operations in quantum theory. We clarify the relationship between maximal correlation and -contraction coefficients by proving they are the same operator norms evaluated on distinct maps. Then we establish new equivalent conditions to the saturation of the data processing inequality for -divergences. This implies previous saturation results for the and sandwiched Rényi divergences. Finally, we establish the quantum maximal correlation coefficients and -contraction coefficients are often efficiently computable. This results in a generic method for efficiently computing mixing times of time-homogeneous quantum Markov chains with a unique full rank fixed point.

Paper Structure

This paper contains 57 sections, 75 theorems, 203 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Overview of Paper and Results
  3. Relation to Previous Work
  4. Petz's Non-Commutative
  5. Preliminaries
  6. The Linear Operator and its Properties
  7. Hadamard Product Representation
  8. Other Properties
  9. Non-Commutative Extensions of
  10. The Choi Isomorphism, Operator Norms, and Hilbert Spaces
  11. Inner Product and Hilbert Spaces
  12. Choi Representation
  13. Connecting Operator Norms and Optimization
  14. Singular Values and Schmidt Decompositions
  15. Quantum Maximal Correlation Coefficients
  16. ...and 42 more sections

Key Result

Proposition 9

Petz-1996a A monotone metric $\gamma_{f,\sigma}(A,B)$ for $\sigma \in \mathrm{D}_{+}$ is monotonic for any map $\mathcal{E}$ that is the adjoint of a unital Schwarz map:

Figures (3)

  • Figure 1: A necessary condition for the existence of a conversion for some integer $n$ under local operations with stochastic maps $\mathcal{W}_{X^{n} \to X'}$ and $\mathcal{R}_{Y^{n} \to Y'}$ is given in terms of the maximal correlation coefficient in the classical setting.
  • Figure 2: Bound on speed of convergence to stationary distribution in the classical setting in terms of maximal correlation coefficie.
  • Figure 3: Summary of the relation between the maximal correlation coefficient and $\chi^{2}$-divergence in (a) classical information theory and (b) how it changes when generalized to quantum theory as established in this work. TVD, TD, DPI, LO, and ACD stand for total variational distance, trace distance, data processing inequality, local operations, and asymptotic common data respectively. A 'strong' monotone means that conversion is limited independent of the number of input copies. If a quantum result includes a citation to a result outside this work, it is because we have obtained an improvement or deeply related result to one in that work.

Theorems & Definitions (162)

  • Definition 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Proposition 9
  • Proposition 10
  • Proposition 11
  • proof
  • Proposition 12
  • proof
  • ...and 152 more