Notes on entropic characteristics of quantum channels

One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss few channel characteristics expressed by means of generalized entropies. Such characteristics can often be dealt in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the $q$-average output entropy of degree $q\geq1$ is bounded from above by the $q$-entropy of the input density matrix. Concavity properties of the $(q,s)$-entropy exchange are considered. Fano type quantum bounds on the $(q,s)$-entropy exchange are derived. We also give upper bounds on the map $(q,s)$-entropies in terms of the output entropy, corresponding to the completely mixed input.Comment: 10 pages, no figures. The statement of Proposition 1 is explicitly illustrated with the depolarizing channel. The bibliography is extended and updated. More explanations. To be published in Cent. Eur. J. Phy

Partitioned trace distances

New quantum distance is introduced as a half-sum of several singular values of difference between two density operators. This is, up to factor, the metric induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar properties to the standard trace distance, including the unitary invariance, the strong convexity and the close relations to the classical distances. The partitioned distances cannot increase under quantum operations of certain kind including bistochastic maps. All the basic properties are re-formulated as majorization relations. Possible applications to quantum information processing are briefly discussed.Comment: 8 pages, no figures. Significant changes are made. New section on majorization is added. Theorem 4.1 is extended. The bibliography is enlarged

Relations for certain symmetric norms and anti-norms before and after partial trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.Comment: 11 pages, no figures. A typo error in Eq. (5.15) is corrected. Minor improvements. J. Stat. Phys. (in press

Global-fidelity limits of state-dependent cloning of mixed states

By relevant modifications, the known global-fidelity limits of state-dependent cloning are extended to mixed quantum states. We assume that the ancilla contains some a priori information about the input state. As it is shown, the obtained results contribute to the stronger no-cloning theorem. An attainability of presented limits is discussed.Comment: 8 pages, ReVTeX, 1 figure. In revised form an attainability of presented limits is discussed. Detected errors are corrected. Elucidative figure is added. Minor grammatical changes are made. More explanation

Continuity and Stability of Partial Entropic Sums

Extensions of Fannes' inequality with partial sums of the Tsallis entropy are obtained for both the classical and quantum cases. The definition of kth partial sum under the prescribed order of terms is given. Basic properties of introduced entropic measures and some applications are discussed. The derived estimates provide a complete characterization of the continuity and stability properties in the refined scale. The results are also reformulated in terms of Uhlmann's partial fidelities.Comment: 9 pages, no figures. Some explanatory and technical improvements are made. The bibliography is extended. Detected errors and typos are correcte