Strong Converse and Stein's Lemma in the Quantum Hypothesis Testing

The hypothesis testing problem of two quantum states is treated. We show a new inequality between the error of the first kind and the second kind, which complements the result of Hiai and Petz to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the first kind error when the power exponent for the second kind error exceeds the quantum relative entropy, and the bound yields the strong converse in the quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi in the classical hypothesis testing.Comment: LaTeX, 12 pages, submitted to IEEE Trans. Inform. Theor

Strong Converse to the Quantum Channel Coding Theorem

A lower bound on the probability of decoding error of quantum communication channel is presented. The strong converse to the quantum channel coding theorem is shown immediately from the lower bound. It is the same as Arimoto's method exept for the difficulty due to non-commutativity.Comment: LaTeX, 11 pages, submitted to IEEE Trans. Inform. Theor

Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions

There are different inequivalent ways to define the R\'enyi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csisz\'ar has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of R\'enyi capacity, defined in terms of the sandwiched quantum R\'enyi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel $W$ with composition $P$ and rate $R$ as $sc(W,R,P)$, our main result is that $$sc(W,R,P)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^*(W,P)\right],$$ where $\chi_{\alpha}^*(W,P)$ is the $P$-weighted sandwiched R\'enyi divergence radius of the image of the channel.Comment: 46 pages. V7: Added the strong converse exponent with cost constrain

Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies

We show that the new quantum extension of Renyi's \alpha-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter \alpha: for \alpha<1, the right choice seems to be the traditional definition, whereas for \alpha>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi \alpha-relative entropies are asymptotically attainable by measurements for \alpha>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.Comment: v5: Added Appendix A on monotonicity and attainability propertie

将来宇宙ミッションに向けたマイクロマシン技術を用いた超軽量X線望遠鏡の研究

首都大学東京, 2016-03-25, 博士（理学）, 甲第565号首都大学東