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A Smooth Entropy Approach to Quantum Hypothesis Testing and the Classical Capacity of Quantum Channels

Datta, Nilanjana and Mosonyi, Milán and Hsieh, Min-Hsiu and Brandão, Fernando G. S. L. (2013) A Smooth Entropy Approach to Quantum Hypothesis Testing and the Classical Capacity of Quantum Channels. IEEE Transactions on Information Theory, 59 (12). pp. 8014-8026. ISSN 0018-9448. doi:10.1109/TIT.2013.2282160.

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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. Then using 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 (ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-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 ε-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.

Item Type:Article
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URLURL TypeDescription Paper
Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:© 2013 IEEE. Manuscript received August 22, 2012; revised May 30, 2013; accepted August 21, 2013. Date of current version November 19, 2013. This work was supported by the European Community’s Seventh Framework Program (FP7/2007–2013) under Grant 213681. M. Mosonyi was supported by the Marie Curie International Incoming Fellowship “QUANTSTAT.” M.-H. Hsieh was supported by the UTS Chancellor’s Postdoctoral Research Fellowship. F. G. S. L. Brandão was supported by the Swiss National Science Foundation, via the National Centre of Competence in Research QSIT. N. Datta would like to thank I. Bjelakovic for a helpful exchange and for pointing out related results for classical and quantum compound channels.
Funding AgencyGrant Number
European Commission213681
Marie Curie FellowshipQUANTSTAT
UTS Chancellor’s Postdoctoral Research FellowshipUNSPECIFIED
Swiss National Science Foundation (SNSF)UNSPECIFIED
Subject Keywords:Capacity, hypothesis testing, quantum channels, smooth max-relative entropy, strong converse
Issue or Number:12
Record Number:CaltechAUTHORS:20160525-154244272
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Official Citation:N. Datta, M. Mosonyi, M. H. Hsieh and F. G. S. L. Brandão, "A Smooth Entropy Approach to Quantum Hypothesis Testing and the Classical Capacity of Quantum Channels," in IEEE Transactions on Information Theory, vol. 59, no. 12, pp. 8014-8026, Dec. 2013. doi: 10.1109/TIT.2013.2282160
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:67368
Deposited By: Tony Diaz
Deposited On:26 May 2016 19:31
Last Modified:11 Nov 2021 00:31

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