Quantum Rényi and $f$-divergences from integral representations

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 $\alpha <1$ and the sandwiched Rényi divergence for $\alpha >1$, 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.

44 pages. v2: improved results on reverse Pinsker inequalities + minor clarifications. v3: some generalizations and clarifications; published version

References in corpus (10)

• The Quantum Chernoff Bound
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Properties of Classical and Quantum Jensen-Shannon Divergence
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Tight constraints on probabilistic convertibility of quantum states
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification
• Continuity of quantum entropic quantities via almost convexity
• Integral formula for quantum relative entropy implies data processing inequality
• Bounding quantum capacities via partial orders and complementarity
• Recoverability of quantum channels via hypothesis testing

Cited by in corpus (7)

• Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing
• Continuity of entropies via integral representations
• Entanglement cost for infinite-dimensional physical systems
• Optimising the relative entropy under semidefinite constraints
• Measured Hockey-Stick Divergence and its Applications to Quantum Pufferfish Privacy
• Quantum Local Differential Privacy and Quantum Statistical Query Model
• Reverse-type Data Processing Inequality