On Composite Quantum Hypothesis Testing

Mario Berta,Christoph Hirche,Fernando G S L Brandão

doi:10.1007/s00220-021-04133-8

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.

Highlights

Hypothesis testing is arguably one of the most fundamental primitives in quantum information theory
A particular hypothesis testing setting is that of quantum state discrimination where quantum states are assigned to each of the hypotheses and we aim to determine which state is given
We are optimizing over all two-outcome positive operator valued measures (POVMs) with {Mn, (1 − Mn)} and

Summary

Overview of Results

Hypothesis testing is arguably one of the most fundamental primitives in quantum information theory. A well studied discrimination setting is that between fixed independent and identical (iid) states ρ⊗n and σ ⊗n, where the asymptotic error exponent is determined by the quantum Stein’s lemma [4,30,43] in terms of the quantum relative entropy. We denote this special case of Eq (5) by ζρ,σ (∞, ε) and the Stein’s lemma gives for any ε ∈ (0, 1) the formula ζρ,σ (∞, ε) = D(ρ σ ) :=.

Proof of Main Result

Examples and Extensions

Conditional Quantum Mutual Information

Conclusion

Some Lemmas