2011/06/30 by Nilanjana Datta, Milan Mosonyi, Min-Hsiu Hsieh +1 · 1 citation
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.1109/tit.2013.2282160
published as IEEE Trans. Inf. Theo. 59, 8014 (2013) · v4: Title changed, improved bounds, both direct and strong converse rates are covered, a new Discussion section added. 20 pages
arxiv created 2013/05/23 · arxiv updated 2013/12/11
We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Using then a relative entropy version of the Quantum Asymptotic Equipartition Property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger (\ep-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot \ep-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the \ep-independent version of the relative entropy-QAEP, we can recover both the Holevo-Schumacher-Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.