On Composite Quantum Hypothesis Testing

Abstract

We extend quantum Stein's lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ${\rho }^{\otimes n}$ against convex combinations of quantum states ${\sigma }^{\otimes n}$ can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein's lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy -- featuring an explicit and universal recovery map.

14 pages, v2: matches published version, includes several corrections

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• Asymptotic relative submajorization of multiple-state boxes
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• Lower Bounds on Error Exponents via a New Quantum Decoder
• Black Box Work Extraction and Composite Hypothesis Testing
• Analytical bounds for non-asymptotic asymmetric state discrimination
• Asymptotic quantification of entanglement with a single copy
• Chain rules for quantum channels
• Distributed Quantum Hypothesis Testing under Zero-rate Communication Constraints
• Limits of Detecting Extraterrestrial Civilizations
• Universal tester for multiple independence testing and classical-quantum arbitrarily varying multiple access channel
• Towards Quantum Universal Hypothesis Testing
• Intersection and union of subspaces with applications to communication over authenticated classical-quantum channels and composite hypothesis testing
• Performance Guarantees for Quantum Neural Estimation of Entropies