This paper reviews Holevo’s contributions to quantum information theory during the 20 century. At that time, he mainly studied three topics, classical-quantum channel coding, quantum estimation with Craméro–Rao approach and quantum estimation with the group covariant approach. This paper addresses these three topics.
Abstract This review aims at gathering the most relevant quantum multi-parameter estimation methods that go beyond the direct use of the quantum Fisher information concept. We discuss in detail the Holevo Cramér–Rao bound, the quantum local asymptotic normality approach as well as Bayesian methods. Even though the fundamental concepts in the field have been laid out more than forty years ago, a number of important results have appeared much more recently. Moreover, the field drew increased attention recently thanks to advances in practical quantum metrology proposals and implementations that often involve estimation of multiple parameters simultaneously. Since the topics covered in these review are spread in the literature and often served in a very formal mathematical language, one of the main goals of this review is to provide a largely self-contained work that allows the reader to follow most of the derivations and get an intuitive understanding of the interrelations between different concepts using a set of simple yet representative examples involving qubit and Gaussian shift models.
Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven. For classical information transmission via quantum channels we give a new proof of the coding theorem, and prove the strong converse, even under the extended model of nonstationary channels. As a by-product we obtain a new proof of the famous Holevo bound. Then multi-user systems are investigated, and the capacity region for the quantum multiple access channel is determined. The last chapter contains a preliminary discussion of some models of compression of correlated quantum sources, and a proposal for a program to obtain operational meaning for quantum conditional entropy. An appendix features the introduction of a notation and calculus of entropy in quantum systems.
This report surveys quantum error-correcting codes. As Preskill claimed, 21st century would be the golden age of quantum error correction. Quantum channels behave differently from classical channels, so researchers face difficulties in developing robust quantum codes. Fortunately, the classical error control methods have been well developed. If we can learn many lessons from classical coding theory, we can expedite the development of quantum codes. Scientists have discovered that quantum error correction shares many concepts with classical counterpart. Both quantum and classical coding schemes add redundancy to information to protect against noises. They also have similar conditions for error detectability and correctability.
The purpose of this review is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review and of more detailed examples—coming mainly from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.