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.

arXiv comment44 pages. v2: improved results on reverse Pinsker inequalities + minor clarifications. v3: some generalizations and clarifications; published version

References in corpus (34)

• On quantum Renyi entropies: a new generalization and some properties
• The Quantum Chernoff Bound
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• $f$-divergence Inequalities
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Asymptotic Error Rates in Quantum Hypothesis Testing
• Properties of Classical and Quantum Jensen-Shannon Divergence
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• Quantum f-divergences and error correction
• Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces
• Quantum logarithmic Sobolev inequalities and rapid mixing
• The ${\chi }^2$-divergence and Mixing times of quantum Markov processes
• Amortized Channel Divergence for Asymptotic Quantum Channel Discrimination
• Continuity bounds on the quantum relative entropy
• Continuity bounds on the quantum relative entropy - II
• Quantum Skew Divergence
• Relative Entropy Convergence for Depolarizing Channels
• Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels
• Optimized quantum f-divergences and data processing
• Investigating Properties of a Family of Quantum Renyi Divergences
• Tight constraints on probabilistic convertibility of quantum states
• On contraction coefficients, partial orders and approximation of capacities for quantum channels
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification
• Optimal Extensions of Resource Measures and their Applications
• Contraction coefficients for noisy quantum channels
• Families of completely positive maps associated with monotone metrics
• Continuity of quantum entropic quantities via almost convexity
• Sandwiched Rényi Convergence for Quantum Evolutions
• 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
• A lower bound on the fidelity between two states in terms of their trace-distance and max-relative entropy
• Upper continuity bound on the quantum quasi-relative entropy

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• 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