Pretty good measures in quantum information theory

Abstract

Quantum generalizations of Renyi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Renyi divergence ${\bar{D}}_{\alpha }$ and the minimal quantum Renyi divergence ${\tilde{D}}_{\alpha }$. In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely that $\alpha {\bar{D}}_{\alpha }(\rho \Vert \sigma )\le {\tilde{D}}_{\alpha }(\rho \Vert \sigma )$ for $\alpha \in [0,1]$ and where $\rho$ and $\sigma$ are density operators. This bound suggests defining a "pretty good fidelity", whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction. We also find a new necessary and sufficient condition for optimality of the pretty good measurement and singlet fraction.

15.1 pages; v2: 16 pages, accepted for publication in IEEE Transactions on Information Theory

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• Amortized Channel Divergence for Asymptotic Quantum Channel Discrimination
• Principles of Quantum Communication Theory: A Modern Approach
• Test-measured Rényi divergences
• Strong converse bounds in quantum network information theory: distributed hypothesis testing and source coding
• Optimality Condition for the Petz Map
• Better bounds on optimal measurement and entanglement recovery, with applications to uncertainty and monogamy relations
• Quantifying the unextendibility of entanglement
• A fidelity measure for quantum states based on the matrix geometric mean
• Pretty good measurement for bosonic Gaussian ensembles
• Fidelity Bounds for Device-Independent Advantage Distillation
• Quantum Chernoff divergence in advantage distillation for quantum key distribution and device-independent quantum key distribution
• Relations between different quantum Rényi divergences