On Composite Quantum Hypothesis Testing

Understand

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.