Some continuity properties of quantum Rényi divergences

Abstract

In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel Rényi $\alpha$-divergences over all $\alpha >1$. We prove the equality of these two threshold values (and therefore the strong converse property for this problem) using a minimax argument based on a newly established continuity property of the sandwiched Rényi divergences. Motivated by this, we give a detailed analysis of the continuity properties of various other quantum (channel) Rényi divergences, which may be of independent interest.

arXiv commentv4: Continuity is studied on more general sets of the form $\rho \le f(\sigma )$ for a large class of functions $f$. 44 pages

References in corpus (28)

• On quantum Renyi entropies: a new generalization and some properties
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• A Fully Quantum Asymptotic Equipartition Property
• Asymptotic Error Rates in Quantum Hypothesis Testing
• Monotonicity of a relative Rényi entropy
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• The quantum capacity of channels with arbitrarily correlated noise
• Quantum f-divergences and error correction
• On the quantum Renyi relative entropies and related capacity formulas
• alpha-z-relative Renyi entropies
• Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• Amortized Channel Divergence for Asymptotic Quantum Channel Discrimination
• Different quantum f-divergences and the reversibility of quantum operations
• Strong converse exponent for classical-quantum channel coding
• Using and reusing coherence to realize quantum processes
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture (New title)
• Investigating Properties of a Family of Quantum Renyi Divergences
• Defining quantum divergences via convex optimization
• Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions
• Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences
• Bounding the forward classical capacity of bipartite quantum channels
• Test-measured Rényi divergences

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• Uhlmann's theorem for relative entropies
• Uhlmann's theorem for measured divergences
• One-shot Multiple Access Channel Simulation
• Log-majorizations between quasi-geometric type means for matrices