Alexander S. Holevo

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.

Statistical decision theory for quantum systems

Quantum statistical decision theory arises in connection with applied problems of optimal detection and processing of quantum signals. In this paper we give a systematic treatment of this theory, based on operator-valued measures. We study the existence problem for optimal measurements and give sufficient and necessary conditions for optimality. The notion of the maximum likelihood measurement is introduced and investigated. The general theory is then applied to the case of Gaussian (quasifree) states of Bose systems, for which optimal measurements of the mean value are found.

Quantum coding theorems

The more than thirty years old issue of the (classical) information capacity of quantum communication channels was dramatically clarified during the last years, when a number of direct quantum coding theorems was discovered. The present paper gives a self contained treatment of the subject, following as much in parallel as possible with classical information theory and, on the other side, stressing profound differences of the quantum case. An emphasis is made on recent results, such as general quantum coding theorems including cases of infinite (possibly continuous) alphabets and constrained inputs, reliability function for pure state channels and quantum Gaussian channel. Several still unsolved problems are briefly outlined.

Quantum systems, channels, information : a mathematical introduction

The subject of this book is theory of quantum system presented from information science perspective. The central role is played by the concept of quantum channel and its entropic and information characteristics. Quantum information theory gives a key to understanding elusive phenomena of quantum world and provides a background for development of experimental techniques that enable measuring and manipulation of individual quantum systems. This is important for the new efficient applications such as quantum computing, communication and cryptography. Research in the field of quantum informatics, including quantum information theory, is in progress in leading scientific centers throughout the world. This book gives an accessible, albeit mathematically rigorous and self-contained introduction to quantum information theory, starting from primary structures and leading to fundamental results and to exiting open problems.

Counterexample to an additivity conjecture for output purity of quantum channels

A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the p-norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for p>4.79. The same example (with p=∞) also disproves a conjecture for the multiplicativity of the injective norm of Hilbert space tensor products.

Ultimate classical communication rates of quantum optical channels

V. Giovannetti, R. Garcı́a-Patrón, N. J. Cerf, and A. S. Holevo NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy QuIC, Ecole Polytechnique de Bruxelles, CP 165, Université Libre de Bruxelles, 1050 Bruxelles, Belgium Max-Planck Institut für Quantenoptik, Hans-Kopfermann Str. 1, D-85748 Garching, Germany Steklov Mathematical Institute, RAS, Moscow (Dated: December 24, 2013)

On entanglement-assisted classical capacity

We give a modified proof of the recent result of C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal concerning entanglement-assisted classical capacity of a quantum channel and discuss the relation between entanglement-assisted and unassisted classical capacities.

Problems in the mathematical theory of quantum communication channels

Abstract In this paper a survey of some basic concepts of the general communication theory including both classical and quantum case is given. Some open questions are formulated and discussed.

A Solution of Gaussian Optimizer Conjecture for Quantum Channels

The long-standing conjectures of the optimality of Gaussian inputs for Gaussian channel and Gaussian additivity are solved for a broad class of covariant or contravariant Bosonic Gaussian channels (which includes in particular thermal, additive classical noise, and amplifier channels) restricting to the class of states with finite second moments. We show that the vacuum is the input state which minimizes the entropy at the output of such channels. This allows us to show also that the classical capacity of these channels (under the input energy constraint) is additive and is achieved by Gaussian encodings.

Covariant measurements and uncertainty relations

Abstract The concept of covariant measurement is a generalization of the well-known notion of imprimitivity system. This concept allows mathematical description of such physical quantities as angle, phase etc. in the framework of quantum theory. In this paper the structure of covariant measurements is described in the cases of finite-dimensional or irreducible representation of the symmetry group. A noncommutative analogue of Hunt-Stein theorem in mathematical statistics is proved, giving a characterization of the most precise measurements. As examples, measurements of angle, phase and orientation are considered. Rigorous uncertainty relations are obtained in these cases.