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arXiv:quant-ph/9809023 · Russian Mathematical Surveys

Quantum coding theorems

Alexander Semenovich Holevo

Published Dec 31, 1998• 230 citations
arXivHTMLPDF

Abstract

Contents I. Introduction II. General considerations § 1. Quantum communication channel § 2. Entropy bound and channel capacity § 3. Formulation of the quantum coding theorem. Weak conversion III. Proof of the direct statement of the coding theorem § 1. Channels with pure signal states § 2. Reliability function § 3. Quantum binary channel § 4. Case of arbitrary states with bounded entropy IV. c-q channels with input constraints § 1. Coding theorem § 2. Gauss channel with one degree of freedom § 3. Classical signal on quantum background noise Bibliography

DOI
10.1070/rm1998v053n06abeh000091

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Contents I. Introduction II. General considerations § 1. Quantum communication channel § 2. Entropy bound and channel capacity § 3. Formulation of the quantum coding theorem. Weak conversion III. Proof of the direct statement of the coding theorem § 1. Channels with pure signal states § 2. Reliability function § 3. Quantum binary channel § 4. Case of arbitrary states with bounded entropy IV. c-q channels with input constraints § 1. Coding theorem § 2. Gauss channel with one degree of freedom § 3. Classical signal on quantum background noise Bibliography
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.
We define classical quantum multiway channels for transmission of classical information, after the previous work by Allahverdyan and Saakian (see Quantum Computing and Quantum Communications (Lecture Notes in Computer Science). Berlin, Germany: Springer-Verlag, vol.1509, 1999). Bounds on the capacity region are derived in a uniform way, which are analogous to the classically known ones, simply replacing Shannon (1961) entropy with von Neumann (1955) entropy. For the single receiver case (multiple-access channel) the elect capacity region is determined. These results are applied to the case of noisy channels, with arbitrary input signal states. A second issue of this work is the presentation of a calculus of quantum information quantities, based on the algebraic formulation of quantum theory.
This chapter unifies all of the channel coding theorems that we have studied in this book. One of the most general information-processing tasks that a sender and receiver can accomplish is to transmit classical and quantum information and generate entanglement with many independent uses of a quantum channel and with the assistance of classical communication, quantum communication, and shared entanglement. The resulting rates for communication are net rates that give the generation rate of a resource less its consumption rate. Since we have three resources, all achievable rates are rate triples ( C, Q, E ) that lie in a three-dimensional capacity region, where C is the net rate of classical communication, Q is the net rate of quantum communication, and E is the net rate of entanglement consumption/generation. The capacity theorem for this general scenario is known as the quantum dynamic capacity theorem, and it is the main theorem that we prove in this chapter. All of the rates given in the channel coding theorems of previous chapters are special points in this three-dimensional capacity region. The proof of the quantum dynamic capacity theorem comes in two parts: the direct coding theorem and the converse theorem. The direct coding theorem demonstrates that the strategy for achieving any point in the three-dimensional capacity region is remarkably simple: we just combine the protocol from Corollary 21.5.3 for entanglement-assisted classical and quantum communication with the three unit protocols of teleportation, super-dense coding, and entanglement distribution.