Quantum Rényi and $f$-divergences from integral representations

TL;DR

The paper introduces a quantum f-divergence D_f(ρ||σ) built from an integral over hockey-stick divergences, unifying a broad class of quantum divergences. Remarkably, regularization of the newly defined quantum Rényi divergences recovers the Petz divergence for α<1 and the sandwiched divergence for α>1, effectively merging two major quantum Rényi families. It also demonstrates contraction-coefficient behavior analogous to the classical case, with collapse for operator-convex f, and derives reverse Pinsker-type inequalities with applications to differential privacy and hypothesis testing. The framework yields wide-ranging operational and mathematical insights, including bounds on amortized channel divergences and criteria for less noisy channel orders, highlighting potential applications in quantum privacy and information processing.

Abstract

Smooth Csiszár $f$-divergences can be expressed as integrals over so-called hockey stick divergences. This motivates a natural quantum generalization in terms of quantum Hockey stick divergences, which we explore here. Using this recipe, the Kullback-Leibler divergence generalises to the Umegaki relative entropy, in the integral form recently found by Frenkel. We find that the Rényi divergences defined via our new quantum $f$-divergences are not additive in general, but that their regularisations surprisingly yield the Petz Rényi divergence for $α< 1$ and the sandwiched Rényi divergence for $α> 1$, unifying these two important families of quantum Rényi divergences. Moreover, we find that the contraction coefficients for the new quantum $f$ divergences collapse for all $f$ that are operator convex, mimicking the classical behaviour and resolving some long-standing conjectures by Lesniewski and Ruskai. We derive various inequalities, including new reverse Pinsker inequalities with applications in differential privacy and explore various other applications of the new divergences.

Figures (2)

• Figure 1: Plot of the different Rényi divergences (left) and Hellinger divergences (right) defined in the main text over $\alpha$ for the states specified above. As expected $\widebar D_\alpha$, $\widetilde{D}_\alpha$ and $D_\alpha$ appear equal for $\alpha=1$, however for other values of $\alpha$ the quantity $D_\alpha$ is distinctly different from the others.
• Figure 2: Plot of the relative entropy and several bounds on it for different states over the parameter $p$. (Left) Example for $d=2$. The states commute and hence our bound in Eq. \ref{['Eq:NewRevPin-0']} equals the relative entropy. Our bound in Eq. \ref{['Eq:NewRevPin-1']} is incomparable to the previous bound in Eq. \ref{['Eq:Aud-RevPinsker']}. (Right) Example for $d=3$. Eq. \ref{['Eq:NewRevPin-0']} gives the closest upper bound and is tight for some values of $p$.

Theorems & Definitions (88)

• Lemma 2.1
• proof
• Theorem 2.2: Theorem 6 in frenkel2022integral
• Corollary 2.3
• proof
• Definition 2.4
• Lemma 2.5
• proof
• Proposition 2.6
• proof