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Reports on Mathematical Physics

Problems in the mathematical theory of quantum communication channels

Alexander Semenovich Holevo

Published Oct 1, 1977• 83 citations
Publisher
DOI
10.1016/0034-4877(77)90010-6

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Abstract The communication of information by means of laser is an important technique. Therefore, it is necessary to construct a communication theory (quantum information theory) including quantum effects of laser action. Ohya, one of the authors, has studied the quantum communication channel and constructed a compound state for systematic formulation of the mutual information in quantum systems. In this paper, (1) we briefly review the mathematical formulation of quantum communication theory and (2) construct a physical model of the optical fiber communication process in order to examine the qualitative properties of the communication channel.
We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young’s inequality for convolutions, and the theorem “Gaussian kernels have only Gaussian maximizers.” Quantum Gaussian channels play a key role in quantum communication theory: they are the quantum counterpart of Gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space. The restriction of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides with the thinning, which is the analog of the rescaling for positive integer random variables. Quantum Gaussian channels provide then a bridge between functional analysis and discrete probability.
The main emphasis of this work is the mathematical theory of quantum channels and their entropic and information characteristics. Quantum information theory is one of the key research areas, since it leads the way to vastly increased computing speeds by using quantum systems to store and process information. Quantum cryptography allows for secure communication of classified information. Research in the field of quantum informatics, including quantum information theory, is in progress in leading scientific centers throughout the world. The past years were marked with impressive progress made by several researchers in solution of some difficult problems, in particular, the additivity of the entropy characteristics of quantum channels. This suggests a need for a book that not only introduces the basic concepts of quantum information theory, but also presents in detail some of the latest achievements.