By Harrison D.M.

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**Example text**

In statistics this is often called Kullback-Leibler divergence. (1) of Chap. 5. , by combining classical Stein’s lemma, they showed that the Quantum relative entropy can be achieved by the optimal second error exponent with the constant constraint. Undoubtedly, their achievement is a great progress in Quantum hypothesis testing. However, Quantum Stein’s lemma was not completed only by the above result because an opposite inequality is required. , Hiai-Petz’s result) is called the direct part while the impossibility part is called the converse part.

2] R. E. Blahut, “Hypothesis testing and information theory,” IEEE Trans. Inform. Theory, 20, 405–417, (1974). E. Blahut, Principles and practice of information theory, Reading, Addison-Wesley, MA, 1987. [4] I. Csisz´ ar, “Information type measures of diﬀerence of probability distributions and indirect observations,” Studia Sci. Math. , 2, 299–318, 1967. [5] A. Dembo and O. Zeitouni, Large deviation techniques and applications, Jones and Bartlett, Boston, 1993. W. Helstrom, Quantum detection and estimation theory, Academic Press, New York, 1976.

When every element is randomly generated by a given probability, the maximum size satisfying the asymptotic weak orthogonality is characterized by the von Neumann entropy, but the maximum satisfying the asymptotic strong orthogonality is characterized by the quantum R´enyi entropy with degree 2. Here, we should remark that as was solved by Hausladen et al. [I-17], the von Neumann entropy equals the capacity of quantum channel coding with pure states, and that the quantum R´enyi entropy with degree 2 coincides with the smaller lower bound of the capacity by Stratonovich and Vantsjan [I-18].