On Composite Quantum Hypothesis Testing
On Composite Quantum Hypothesis Testing
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DOI：
10.1007/s00220-021-04133-8
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发表时间：
2017-09
影响因子：
2.4
通讯作者：
M. Berta;F. Brandão;C. Hirche
中科院分区：
文献类型：
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作者：
M. Berta;F. Brandão;C. Hirche
我们将量子Stein引理在非对称量子假设检验中推广到复合零假设和备择假设。作为我们的主要结果，我们证明了检验凸量子态组合对凸量子态组合的渐近误差指数可以写成正则化的量子相对熵公式。我们证明，在一般情况下，这样的正则化是必要的，但也讨论了各种设置，我们的公式以及其扩展成为单字母。这包括在假设检验方面对相干性的相对熵的操作性解释。对于我们的证明，我们从经典概率分布的复合Stein引理开始，并通过使用量子熵的基本性质将结果提升到非对易设置。最后，我们的研究结果也意味着一个改进的可恢复性下界的条件量子互信息的正则化量子相对熵，具有明确的和普遍的恢复地图。
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 statesagainst convex combinations of quantum statescan 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.