Quantum Rényi divergences and the strong converse exponent of state discrimination in operator algebras

Abstract

The sandwiched Rényi $\alpha$-divergences of two finite-dimensional quantum states play a distinguished role among the many quantum versions of Rényi divergences as the tight quantifiers of the trade-off between the two error probabilities in the strong converse domain of state discrimination. In this paper we show the same for the sandwiched Rényi divergences of two normal states on an injective von Neumann algebra, thereby establishing the operational significance of these quantities. Moreover, we show that in this setting, again similarly to the finite-dimensional case, the sandwiched Rényi divergences coincide with the regularized measured Rényi divergences, another distinctive feature of the former quantities. Our main tool is an approximation theorem (martingale convergence) for the sandwiched Rényi divergences, which may be used for the extension of various further results from the finite-dimensional to the von Neumann algebra setting. We also initiate the study of the sandwiched Rényi divergences of pairs of states on a $C^{\ast }$-algebra, and show that the above operational interpretation, as well as the equality to the regularized measured Rényi divergence, holds more generally for pairs of states on a nuclear $C^{\ast }$-algebra.

arXiv comment40 pages. To appear in Annales Henri Poincare

References in corpus (31)

• 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
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• 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
• On the quantum Renyi relative entropies and related capacity formulas
• alpha-z-relative Renyi entropies
• Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• Different quantum f-divergences and the reversibility of quantum operations
• Strong converse exponent for classical-quantum channel coding
• Multivariate Trace Inequalities
• On Variational Expressions for Quantum Relative Entropies
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics
• Error exponents in hypothesis testing for correlated states on a spin chain
• Rényi divergences as weighted non-commutative vector valued $L_p$-spaces
• Rényi relative entropies and noncommutative $L_p$-spaces
• An information-theoretic treatment of quantum dichotomies
• Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions
• Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences
• Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences
• Rényi relative entropies and noncommutative $L_p$-spaces II
• Two approaches to obtain the strong converse exponent of quantum hypothesis testing for general sequences of quantum states
• Test-measured Rényi divergences
• The strong converse exponent of discriminating infinite-dimensional quantum states
• Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings

Cited by in corpus (4)

• The strong converse exponent of discriminating infinite-dimensional quantum states
• Unified framework for continuity of sandwiched Rényi divergences
• Asymptotic Equipartition Theorems in von Neumann algebras
• Super-exponential distinguishability of correlated quantum states