There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution. In [IEEE Transactions on Information Theory, 41(1):26-34, 1995], Csiszár 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 Rényi capacity, defined in terms of the sandwiched quantum Rényi divergences, has the same operational interpretation in the strong converse problem of constant composition classical-quantum channel coding.

We present two general approaches to obtain the strong converse exponent of simple quantum hypothesis testing for correlated quantum states. One approach requires that the states satisfy a certain factorization property; typical examples of such states are the temperature states of translation-invariant finite-range interactions on a spin chain. The other approach requires the differentiability of a regularized Rényi α-divergence in the parameter α; typical examples of such states include temperature states of non-interacting fermionic lattice systems, and classical irreducible Markov chains. In all cases, we get that the strong converse exponent is equal to the Hoeffding antidivergence, which in turn is obtained from the regularized Rényi divergences of the two states.

It is known that a message can be transmitted safely against any wiretapper via a noisy channel without a secret key if the coding rate is less than the so-called secrecy capacity CS, which is usually smaller than the channel capacity C. In order to remove the loss C-CS, we propose a multiplex coding scheme with plural independent messages. In this paper, it is shown that the proposed multiplex coding scheme can attain the channel capacity as the total rate of the plural messages and the perfect secrecy for each massage. Several bounds of achievable multiplex coding rate region are derived for general wiretap channels in the sense of information-spectral methods, by extending Hayashi's proof, in which the coding of the channel resolvability is applied to wiretap channels. Furthermore, the exact region for deterministic coding is determined for stationary memoryless full-rank wiretap channels.

In this correspondence, we give an alternative proof of the direct part of the classical-quantum channel coding theorem (the Holevo-Schumacher-Westmoreland (HSW) theorem), using ideas of quantum hypothesis testing. In order to show the existence of good codes, we invoke a limit theorem, relevant to the quantum Stein's lemma, in quantum hypothesis testing as the law of large numbers used in the classical case. We also apply a greedy construction of good codes using a packing procedure of noncommutative operators. Consequently we derive an upper bound on the coding error probability, which is used to give an alternative proof of the HSW theorem. This approach elucidates how the Holevo information applies to the classical-quantum channel coding problems

In the simple quantum hypothesis testing problem for two density operators, upper bounds on the error probabilities are shown based on a key operator inequality between a density operator and a conditional expectation of it. Concerning the error exponents, the upper bounds lead to a noncommutative analog of the Hoeffding bound, which is identical with the classical counterpart if two density operators commute. The upper bounds also provide a simple proof of the direct part of the quantum Stein's lemma.

The hypothesis testing problem for two quantum states is treated. We show a new inequality between the errors of the first kind and the second kind, which complements the result of Hiai and Petz (1991) to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy, which yields the strong converse in quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi (1989) in classical hypothesis testing.

A lower bound on the probability of decoding error for a quantum communication channel is presented, from which the strong converse to the quantum channel coding theorem is immediately shown. The results and their derivations are mostly straightforward extensions of the classical counterparts which were established by Arimoto (1973), except that more careful treatment is necessary here due to the noncommutativity of operators.