A Smooth Entropy Approach to Quantum Hypothesis Testing and the Classical Capacity of Quantum Channels
Abstract
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.
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.Attached Files
Submitted - 1106.3089v4.pdf
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Additional details
- Eprint ID
- 67368
- DOI
- 10.1109/TIT.2013.2282160
- Resolver ID
- CaltechAUTHORS:20160525-154244272
- 213681
- European Commission
- QUANTSTAT
- Marie Curie Fellowship
- UTS Chancellor's Postdoctoral Research Fellowship
- Swiss National Science Foundation (SNSF)
- Created
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2016-05-26Created from EPrint's datestamp field
- Updated
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2021-11-11Created from EPrint's last_modified field