We show finite-size bounds on the deviation of the optimal type II error from its asymptotic value in the quantum hypothesis testing problem of Stein's lemma with composite null-hypothesis. The proof is based on some simple properties of a new notion of quantum Rènyi divergence, recently introduced in [Müller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013)], and [Wilde, Winter, Yang, arXiv:1306.1586].
@InProceedings{mosonyi:LIPIcs.TQC.2014.88, author = {Mosonyi, Mil\'{a}n}, title = {{Convexity Properties of the Quantum R\'{e}nyi Divergences, with Applications to the Quantum Stein's Lemma}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {88--98}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.88}, URN = {urn:nbn:de:0030-drops-48094}, doi = {10.4230/LIPIcs.TQC.2014.88}, annote = {Keywords: Quantum R\'{e}nyi divergences, Stein's lemma, composite null-hypothesis, second-order asymptotics} }
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