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Information theoretic parameters of non-commutative graphs and convex corners

Gareth Boreland, Ivan G. Todorov, Andreas Winter

TL;DR

The paper develops a comprehensive quantum-information-theoretic framework for non-commutative graphs via convex $M_d$-corners and their anti-blockers. It proves a second anti-blocker theorem, establishes continuity properties, and defines entropy-like optimization parameters $H_{\mathcal{A}}(\rho)$ that generalise classical graph entropy and fractional chromatic numbers. The authors connect these constructs to canonical corners arising from non-commutative graphs, derive a quantum entropy duality $\max_{\rho}H(\mathcal{S},\rho)=\log\chi_f(\mathcal{S})$, and introduce the Witsenhausen rate for non-commutative graphs, along with multiplicativity and tensor-product behavior. The results unify classical and quantum zero-error information theory, extend key combinatorial quantities to the quantum setting, and provide a suite of tools and examples to study information-processing tasks under quantum constraints. Collectively, this work lays a solid foundation for analyzing entropy, chromatic-type quantities, and rate limits in quantum confusability structures.

Abstract

We establish a second anti-blocker theorem for non-commutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimisation parameters for a given convex corner that generalise well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterise its maximum value in terms of a generalised fractional chromatic number and establish entropy splitting results that demonstrate the entropic complementarity between a convex corner and its anti-blocker. We identify two extremal tensor products of convex corners and examine the behaviour of the introduced parameters with respect to tensoring. Specialising to non-commutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of non-commutative graph entropy of a state, which we show to be continuous with respect to the state and the graph. We define the Witsenhausen rate of a non-commutative graph and compute the values of our parameters in some specific cases.

Information theoretic parameters of non-commutative graphs and convex corners

TL;DR

The paper develops a comprehensive quantum-information-theoretic framework for non-commutative graphs via convex -corners and their anti-blockers. It proves a second anti-blocker theorem, establishes continuity properties, and defines entropy-like optimization parameters that generalise classical graph entropy and fractional chromatic numbers. The authors connect these constructs to canonical corners arising from non-commutative graphs, derive a quantum entropy duality , and introduce the Witsenhausen rate for non-commutative graphs, along with multiplicativity and tensor-product behavior. The results unify classical and quantum zero-error information theory, extend key combinatorial quantities to the quantum setting, and provide a suite of tools and examples to study information-processing tasks under quantum constraints. Collectively, this work lays a solid foundation for analyzing entropy, chromatic-type quantities, and rate limits in quantum confusability structures.

Abstract

We establish a second anti-blocker theorem for non-commutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimisation parameters for a given convex corner that generalise well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterise its maximum value in terms of a generalised fractional chromatic number and establish entropy splitting results that demonstrate the entropic complementarity between a convex corner and its anti-blocker. We identify two extremal tensor products of convex corners and examine the behaviour of the introduced parameters with respect to tensoring. Specialising to non-commutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of non-commutative graph entropy of a state, which we show to be continuous with respect to the state and the graph. We define the Witsenhausen rate of a non-commutative graph and compute the values of our parameters in some specific cases.

Paper Structure

This paper contains 25 sections, 75 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Convex $\mathbb{R}^d$-corners
  4. Convex corners and anti-blockers in $M_d$
  5. Definitions and basic properties
  6. Examples of convex $M_d$-corners
  7. Reflexivity of convex $M_d$-corners
  8. The second anti-blocker theorem
  9. Consequences of reflexivity
  10. Non-commutative lifts
  11. Entropy with respect to a convex corner
  12. Background
  13. Quantisation of entropy
  14. Dependence on the state and on the convex corner
  15. Entropy splitting
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $\mathcal{A}$ be a convex $\mathbb{R}^d$-corner. The following are equivalent:

Theorems & Definitions (158)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 148 more