Complementarity and Additivity for Covariant Channels

This paper contains several new results concerning covariant quantum channels in d ≥ 2 dimensions. The first part, Sec. 3, based on [4], is devoted to unitarily covariant channels, namely depolarizing and transpose-depolarizing channels. The second part, Sec. 4, based on [10], studies Weyl-covariant channels. These results are preceded by Sec. 2 in which we discuss various representations of general completely positive maps and channels. In the first part of the paper we compute complementary channels for depolarizing and transpose-depolarizing channels. This method easily yields minimal Kraus representations from non-minimal ones. We also study properties of the output purity of the tensor product of a channel and its complementary. In the second part, the formalism of discrete noncommutative Fourier transform is developed and applied to the study of Weyl-covariant maps and channels. We then extend a result in [16] concerning a bound for the maximal output 2-norm of a Weyl-covariant channel. A class of maps which attain the bound is introduced, for which the multiplicativity of the maximal output 2-norm is proven. The complementary channels are described which have the same multiplicativity properties as the Weyl-covariant channels.     

On States,Channels, and Purification

In this note we introduce purification for a pair (, ), where is a quantum state and is a channel, which allows in particular a natural extension of the properties of related information quantities (mutual and coherent informations) to the channels with arbitrary input and output spaces. PACS: 03.67.Hk     

On classical capacities of infinite-dimensional quantum channels

A coding theorem for entanglement-assisted communication via an infinite-dimensional quantum channel with linear constraints is extended to a natural degree of generality. Relations between the entanglement-assisted classical capacity and χ-capacity of constrained channels are obtained, and conditions for their coincidence are given. Sufficient conditions for continuity of the entanglement-assisted classical capacity as a function of a channel are obtained. Some applications of the obtained results to analysis of Gaussian channels are considered. A general (continuous) version of the fundamental relation between coherent information and the measure of privacy of classical information transmission via an infinite-dimensional quantum channel is proved.     

Locating a photon: Position observables and an uncertainty relation

This paper is written in celebration of Kamil A. Valiev’s 75th birthday and 50-year career. His wide-ranging work includes running workshops on quantum information science at the Academy’s Institute of Physics and Technology, events that invariably attract experts and beginners alike. He often gives thought-provoking talks in which issues of quantum mechanics are considered from an unusual angle. A series of lectures on the theory of the photon provided a basis for the present theoretical study dealing with the long-standing question of how to locate a photon. This topic has acquired a new dimension with the task of manipulating photons, individually or collectively, for the purposes of quantum cryptography.     

Gaussian classical-quantum channels: Gain from entanglement-assistance

In the present paper we introduce and study Bosonic Gaussian classical-quantum (c-q) channels; embedding of the classical input in quantum input is always possible, and therefore the classical entanglement-assisted capacity C _ea under an appropriate input constraint is well defined. We prove the general property of entropy increase for a weak complementary channel, which implies the equality C = C _ea (where C is the unassisted capacity) for a certain class of c-q Gaussian channels under an appropriate energy-type constraint. On the other hand, we show by an explicit example that the inequality C < C _ea is not unusual for constrained c-q Gaussian channel.     

One-mode quantum Gaussian channels: Structure and quantum capacity

A complete classification of one-mode Gaussian channels is given up to canonical unitary equivalence. We also comment on the quantum capacity of these channels. A channel complementary to the quantum channel with additive classical Gaussian noise is described, providing an example of a one-mode Gaussian channel which is neither degradable nor antidegradable.     

On a Sufficient Condition for Additivity in Quantum Information Theory

In this paper we give a sufficient condition for additivity of the minimum output entropy for a pair of given channels and an analytic verification of this condition for specific quantum channels breaking a closely related multiplicativity property [1, 2]. This yields validity of the additivity conjecture for these channels, a result obtained by a different method in [3]. Our proof relies heavily upon certain concavity properties of the output entropy, which are of independent interest.__________Translated from Problemy Peredachi Informatsii, No. 2, 2005, pp. 9–25.Original Russian Text Copyright © 2005 by Datta, Holevo, Suhov.This work was initiated when the second author was an overseas visiting scholar at St John’s College, Cambridge. Afterwards he was supported by the Russian Scientific School Program, project no. 1758.2003.1. The first and third authors worked in association with the CMI, University of Cambridge - MIT.     

Information capacity of a quantum observable

In this paper we consider the classical capacities of quantum-classical channels corresponding to measurement of observables. Special attention is paid to the case of continuous observables. We give formulas for unassisted and entanglement-assisted classical capacities C and C _ea and consider some explicitly solvable cases, which give simple examples of entanglement-breaking channels with C < C _ea. We also elaborate on the ensemble-observable duality to show that C _ea for the measurement channel is related to the χ-quantity for the dual ensemble in the same way as C is related to the accessible information. This provides both accessible information and the χ-quantity for quantum ensembles dual to our examples.     

The capacity of the quantum channel with general signal states

It is shown that the capacity of a classical-quantum channel with arbitrary (possibly mixed) states equals the maximum of the entropy bound with respect to all a priori distributions. This completes the recent result of Hausladen, Jozsa, Schumacher, Westmoreland, and Wootters (1996), who proved the equality for the pure state channel