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 maxrelative 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 entropyQAEP. Similarly, we provide bounds on the oneshot $\ep$error classical capacity of a quantum channel in terms of a smooth maxrelative entropy variant of its Holevo capacity. Using these bounds and the $\ep$independent version of the relative entropyQAEP, we can recover both the HolevoSchumacherWestmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.
 Publication:

arXiv eprints
 Pub Date:
 June 2011
 arXiv:
 arXiv:1106.3089
 Bibcode:
 2011arXiv1106.3089D
 Keywords:

 Quantum Physics
 EPrint:
 v4: Title changed, improved bounds, both direct and strong converse rates are covered, a new Discussion section added. 20 pages