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Metric Properties: From $S$-Divergence to Quantum Jensen Divergence

Teng Zhang

TL;DR

The paper develops a unified trace-log geometric framework for divergences in infinite-dimensional and operator-algebraic settings. By extending the S-divergence and the quantum Jensen–Shannon divergence to tracial $C^*$-algebras and finite von Neumann algebras, it proves that the square roots of these divergences yield metrics on appropriate positive cones and states, and it provides integral representations linking QJSD to shifted trace-log distances. A general mechanism is established for Jensen $f$-divergences with operator convex generators, giving metricity via a Nevanlinna–Stieltjes representation. The work further shows that, although these constructions are robust in the von Neumann/trace framework and extend to GNS envelopes, the QJSD metric is not Hilbertian in general (including higher-dimensional density matrices), highlighting intrinsic geometric limits for Hilbert-space embeddings in quantum information geometry. Collectively, the results deepen the metric structure of quantum divergences and extend their applicability to infinite-dimensional, tracial, and noncommutative settings while clarifying embedding limitations.

Abstract

We extend the trace-logarithmic $S$-divergence from matrices to tracial $C^*$-algebras and finite von Neumann algebras, and show that its square root defines a metric on the invertible positive cone. We also prove an integral representation of the quantum Jensen--Shannon divergence in terms of shifted trace-log distances, implying metricity of its square root on the full positive cone in the same tracial framework. In the matrix case, we answer two questions of Virosztek \cite{Vir21} on Hilbertianity. Finally, we show that symmetric quantum Jensen divergences generated by non-affine operator convex functions yield metrics in the tracial setting via a Nevanlinna--Stieltjes type representation of the derivative, which generalizes a result of Carlen, Lieb and Seiringer.

Metric Properties: From $S$-Divergence to Quantum Jensen Divergence

TL;DR

The paper develops a unified trace-log geometric framework for divergences in infinite-dimensional and operator-algebraic settings. By extending the S-divergence and the quantum Jensen–Shannon divergence to tracial -algebras and finite von Neumann algebras, it proves that the square roots of these divergences yield metrics on appropriate positive cones and states, and it provides integral representations linking QJSD to shifted trace-log distances. A general mechanism is established for Jensen -divergences with operator convex generators, giving metricity via a Nevanlinna–Stieltjes representation. The work further shows that, although these constructions are robust in the von Neumann/trace framework and extend to GNS envelopes, the QJSD metric is not Hilbertian in general (including higher-dimensional density matrices), highlighting intrinsic geometric limits for Hilbert-space embeddings in quantum information geometry. Collectively, the results deepen the metric structure of quantum divergences and extend their applicability to infinite-dimensional, tracial, and noncommutative settings while clarifying embedding limitations.

Abstract

We extend the trace-logarithmic -divergence from matrices to tracial -algebras and finite von Neumann algebras, and show that its square root defines a metric on the invertible positive cone. We also prove an integral representation of the quantum Jensen--Shannon divergence in terms of shifted trace-log distances, implying metricity of its square root on the full positive cone in the same tracial framework. In the matrix case, we answer two questions of Virosztek \cite{Vir21} on Hilbertianity. Finally, we show that symmetric quantum Jensen divergences generated by non-affine operator convex functions yield metrics in the tracial setting via a Nevanlinna--Stieltjes type representation of the derivative, which generalizes a result of Carlen, Lieb and Seiringer.
Paper Structure (31 sections, 30 theorems, 136 equations)

This paper contains 31 sections, 30 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. The $S$-divergence
  3. Quantum Jensen--Shannon divergence
  4. Quantum Jensen $f$-divergences and trace Jensen gaps
  5. Von Neumann preliminaries: determinants and rearrangements
  6. Fuglede--Kadison determinant
  7. Generalized $s$-numbers and a trace formula
  8. The trace-log distance on $\mathcal{M}_{++}$
  9. Arithmetic and geometric means
  10. Equivalent forms of the distance
  11. Basic metric axioms (except triangle inequality)
  12. Congruence invariance and scaling
  13. Scalar divergence as a Hilbert space distance
  14. A rearrangement lower bound for tau(log(X+Y))
  15. Triangle inequality for the trace-log distance on M++
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $A,B\in M_n^{++}(\mathbb{C})$. Define $\delta_S$ by eq:sra. Then $\delta_S$ is a metric on $M_n^{++}(\mathbb{C})$. Moreover, it does not admit any Hilbert space embedding for any $n\ge 2$.

Theorems & Definitions (63)

  • Theorem 1.1: Sra
  • Theorem 1.2
  • Theorem 1.3: Virosztek
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: Carlen--Lieb--Seiringer
  • Theorem 1.9
  • Corollary 1.10
  • Remark 2.1: Normalization of the trace
  • Remark 2.2: Endpoint behavior of $\mu_X$
  • ...and 53 more