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From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence

Dimitri Lanier, Julien Béguinot, Olivier Rioul

TL;DR

This work addresses how to lift classical $f$-divergence inequalities to the quantum domain by using explicit classical distributions that realize maximal $f$-divergence. It proves Matsumoto's maximal divergence theorem with an explicit CPTP channel and distributions so that $D_f^{\max}(\rho\|\sigma)=D_f(r\|s)$, establishing a classical-to-quantum bridge via data processing. The authors derive a data-processing inequality for $D_f^{\max}$ and show that any classical inequality $D_f(p\|q)\le \phi(D_g(p\|q))$ extends to the quantum setting as $D_f(\rho\|\sigma)\le \phi(D_g^{\max}(\rho\|\sigma))$. They apply this framework to obtain an improved quantum Pinsker bound for $\chi^2$-divergence and universal reverse Pinsker inequalities for general $f$-divergences, with comparisons to existing bounds, emphasizing a simple, matrix-free methodology with broad applicability.

Abstract

Explicit classical states achieving maximal $f$-divergence are given, allowing for a simple proof of Matsumoto's Theorem, and the systematic extension of any inequality between classical $f$-divergences to quantum $f$-divergences. Our methodology is particularly simple as it does not require any elaborate matrix analysis machinery but only basic linear algebra. It is also effective, as illustrated by two examples improving existing bounds: (i)~an improved quantum Pinsker inequality is derived between $χ^2$ and trace norm, and leveraged to improve a bound in decoherence theory; (ii)~a new reverse quantum Pinsker inequality is derived for any quantum $f$-divergence, and compared to previous (Audenaert-Eisert and Hirche-Tomamichel) bounds.

From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence

TL;DR

This work addresses how to lift classical -divergence inequalities to the quantum domain by using explicit classical distributions that realize maximal -divergence. It proves Matsumoto's maximal divergence theorem with an explicit CPTP channel and distributions so that , establishing a classical-to-quantum bridge via data processing. The authors derive a data-processing inequality for and show that any classical inequality extends to the quantum setting as . They apply this framework to obtain an improved quantum Pinsker bound for -divergence and universal reverse Pinsker inequalities for general -divergences, with comparisons to existing bounds, emphasizing a simple, matrix-free methodology with broad applicability.

Abstract

Explicit classical states achieving maximal -divergence are given, allowing for a simple proof of Matsumoto's Theorem, and the systematic extension of any inequality between classical -divergences to quantum -divergences. Our methodology is particularly simple as it does not require any elaborate matrix analysis machinery but only basic linear algebra. It is also effective, as illustrated by two examples improving existing bounds: (i)~an improved quantum Pinsker inequality is derived between and trace norm, and leveraged to improve a bound in decoherence theory; (ii)~a new reverse quantum Pinsker inequality is derived for any quantum -divergence, and compared to previous (Audenaert-Eisert and Hirche-Tomamichel) bounds.
Paper Structure (15 sections, 7 theorems, 59 equations, 2 figures)

This paper contains 15 sections, 7 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Explicit Maximal Divergence Theorem
  3. Derivation
  4. Alternative Expressions and a Few Examples
  5. $f(x)=x\log(x)$
  6. $f(x)=x^2-1$ (or $(x-1)^2$)
  7. $f(x)=|x-1|$
  8. Data Processing Inequality for Maximal $f$-Divergence
  9. Application: From Classical to Quantum
  10. Applications to (Reverse) Pinkser Inequalities
  11. Improved Quantum Pinsker Inequality for $\chi^2$-divergence
  12. Reverse Pinsker Inequalities for $f$-Divergences
  13. Comparison to Hirche-Tomamichel's Bound
  14. Comparison to Audenaert-Eisert's Bound
  15. Conclusion

Key Result

Theorem 1

is maximal over all quantum $f$-divergences $D_f(\rho\|\sigma)$: for any states $\rho,\sigma$ (where state $\sigma$ is invertible). Furthermore, there exist classical distributions $r,s$ depending only on $\rho,\sigma$ (explicit expressions eq:expressions given below in the proof) such that

Figures (2)

  • Figure 1: Comparison of the previous Temme et al. bound and the improved bound of this paper on the trace norm distance $||\rho_t - \sigma||_1$ as a function of time, when $\lambda = 0.1$. Different curves correspond to different initial values of the chi-squared divergence $\chi^2(\rho_0 || \sigma)$. The improved bound provides a tighter estimate at earlier times, particularly for larger initial $\chi^2(\rho_0 || \sigma)$.
  • Figure 2: Scatter plot comparing our bound \ref{['eq:QuantumBinette']} to Audenaert-Eisert's bound \ref{['eq:AEbound']} on 10,000 random positive $4\times4$ matrices of unit trace. Points below the diagonal red line are instances where our bound \ref{['eq:QuantumBinette']} is tighter.

Theorems & Definitions (16)

  • Definition 1: Classical $f$-Divergence Csiszar1967
  • Definition 2: Quantum $f$-Divergence
  • Theorem 1
  • proof
  • Theorem 2: Data Processing Inequalities
  • proof : Proof of Theorem \ref{['thm:dpi']}
  • Corollary 1: Classical to Quantum
  • proof
  • Theorem 3: Improved Quantum Pinsker Inequality for $\chi^2$-Divergence
  • proof
  • ...and 6 more