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Mathematical Notes

Quantum proper entropies and additive capacities of quantum information channels

V. P. Belavkin

Published Feb 1, 2011• 0 citations
Publisher

Abstract

An elementary algebraic approach to unified quantum information theory is given. The operational meaning of entanglement as specifically quantum encoding is disclosed. General relative entropy as information divergence is introduced, and three most important types of relative information, namely, the Araki-Umegaki type (A-type), the Belavkin-Staszewski type (B-type), and the thermodynamical (C-type) are discussed. It is shown that true quantum entanglement-assisted entropy is greater than semiclassical (von Neumann) quantum entropy, and the proper positive quantum conditional entropy is introduced. The general quantum mutual information via entanglement is defined, and the corresponding types of quantum channel capacities as a supremum via the generalized encodings are formulated. The additivity problem for quantum logarithmic capacities for products of arbitrary quantum channels under appropriate constraints on encodings is discussed. It is proved that true quantum capacity, which is achieved on the standard entanglement as an optimal quantum encoding, retains the additivity property of the logarithmic quantum channel entanglement-assisted capacities on the products of quantum input states. This result for quantum logarithmic information of A-type, which was obtained earlier by the author, is extended to any type of quantum information.

DOI
10.1134/s0001434611010020

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An elementary algebraic approach to unified quantum information theory is given. The operational meaning of entanglement as specifically quantum encoding is disclosed. General relative entropy as information divergence is introduced, and three most important types of relative information, namely, the Araki-Umegaki type (A-type), the Belavkin-Staszewski type (B-type), and the thermodynamical (C-type) are discussed. It is shown that true quantum entanglement-assisted entropy is greater than semiclassical (von Neumann) quantum entropy, and the proper positive quantum conditional entropy is introduced. The general quantum mutual information via entanglement is defined, and the corresponding types of quantum channel capacities as a supremum via the generalized encodings are formulated. The additivity problem for quantum logarithmic capacities for products of arbitrary quantum channels under appropriate constraints on encodings is discussed. It is proved that true quantum capacity, which is achieved on the standard entanglement as an optimal quantum encoding, retains the additivity property of the logarithmic quantum channel entanglement-assisted capacities on the products of quantum input states. This result for quantum logarithmic information of A-type, which was obtained earlier by the author, is extended to any type of quantum information.
An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki-Umegaki (a-type) and of Belavkin-Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of V. P. Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.
An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki–Umegaki (a-type) and of Belavkin–Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.
Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.
von Neumann entropy represents the amount of information in the quantum state, and this was extended by Ohya for general quantum systems [10]. Umegaki first defined the quantum relative entropy for -finite von Neumann algebras, which was extended by Araki, and Uhlmann, for general von Neumann algebras and *-algebras, respectively. In 1983 Ohya introduced the quantum mutual entropy by using compound states; this describes the amount of information correctly transmitted through the quantum channel, which was also extended by Ohya for general quantum systems. In this paper, we briefly explain Ohya?s -mixing entropy and the quantum mutual entropy for general quantum systems. By using structure equivalent class, we will introduce entropy type functionals based on quantum information theory to improve treatment for the Gaussian communication process.
Capacities of quantum channels are fundamental quantities in the theory of quantum information. A desirable property is the additivity for a capacity measure. However, this cannot be achieved for a few quantities that have been established as capacity measures. Asymptotic regularization is generically necessary making the study of capacities notoriously hard. In this work, by a proper refinement of the physical settings of quantum communication using restricted encodings, we prove additive quantities for quantum channel capacities that can be employed for quantum Shannon theorems. This refinement is consistent with the principles of quantum theory, and it further demonstrates von Neumann entropy as the cornerstone of quantum information.