Algebraic Quantum Information Theory

The theory of quantum communication and technology have been one of hot research topics in information science and technology since 1980's. In this paper, we survey some research development of several mathematical problems in quantum information theory. Particularly, we focus on the important role of combinatorics (including graph theory), number theory, algebra and algebraic geometry (the arithmetic theory of algebraic curves over finite field) in investigating quantum measurement and quantum error-correction.The theory of quantum communication and technology have been one of hot research topics in information science and technology since 1980's. In this paper, we survey some research development of several mathematical problems in quantum information theory. Particularly, we focus on the important role of combinatorics (including graph theory), number theory, algebra and algebraic geometry (the arithmetic theory of algebraic curves over finite field) in investigating quantum measurement and quantum error-correction.

Scientia Sinica MathematicaApr 18, 20172 citations

The mathematics of quantum information theory

Guillaume Aubrun, Stanisław Szarek

Mathematical surveys and monographsAug 30, 20170 citations

Quantum Information Theory

János A. Bergou, Mark S. Hillery, Mark E. Saffman

Graduate texts in physicsJan 1, 20211 citation

Hypothesis testing and Stein's lemma in general probability theories with Euclidean Jordan algebra and its quantum realization

Kanta Sonoda, Hayato Arai, Masahito Hayashi

Even though quantum information theory gives advantage over classical information theory, these two information theories have a structural similarity that many exponet rates of information tasks asymptotically equal to entropic quantities. A typical example is Stein's Lemma, which many researchers still keep interested in. In this paper, in order to analyze the mathemtaical roots of the structural similarity, we investigate mathematically minimum structure where Stein's Lemma holds. We focus on the structure of Euclidean Jordan Algebras (EJAs), which is a generalization of the algebraic structure in quantum theory, and we investigate the properties of general models of General Probabilistic Theories (GPTs) generated by EJAs. As a result, we prove Stein's Lemma in any model of GPTs generated by EJAs by establishing a generalization of information theoretical tools from the mathematical properties of EJAs.Even though quantum information theory gives advantage over classical information theory, these two information theories have a structural similarity that many exponet rates of information tasks asymptotically equal to entropic quantities. A typical example is Stein's Lemma, which many researchers still keep interested in. In this paper, in order to analyze the mathemtaical roots of the structural similarity, we investigate mathematically minimum structure where Stein's Lemma holds. We focus on the structure of Euclidean Jordan Algebras (EJAs), which is a generalization of the algebraic structure in quantum theory, and we investigate the properties of general models of General Probabilistic Theories (GPTs) generated by EJAs. As a result, we prove Stein's Lemma in any model of GPTs generated by EJAs by establishing a generalization of information theoretical tools from the mathematical properties of EJAs.

arXiv:2505.02487May 5, 20250 citations

Algebraic approach to quantum theory: a finite-dimensional guide

Cédric Bény, Richter, Florian

This document is meant as a pedagogical introduction to the modern language used to talk about quantum theory, especially in the field of quantum information. It assumes that the reader has taken a first traditional course on quantum mechanics, and is familiar with the concept of Hilbert space and elementary linear algebra. As in the popular textbook on quantum information by Nielsen and Chuang, we introduce the generalised concept of states (density matrices), observables (POVMs) and transformations (channels), but we also characterise these structures from an algebraic standpoint, which provides many useful technical tools, and clarity as to their generality. This approach also makes it manifest that quantum theory is a direct generalisation of probability theory, and provides a unifying formalism for both fields. The focus on finite-dimensional systems allows for a self-contained presentation which avoids many of the technicalities inherent to the more general $C^*$ -algebraic approach, while being appropriate for the quantum information literature.This document is meant as a pedagogical introduction to the modern language used to talk about quantum theory, especially in the field of quantum information. It assumes that the reader has taken a first traditional course on quantum mechanics, and is familiar with the concept of Hilbert space and elementary linear algebra. As in the popular textbook on quantum information by Nielsen and Chuang, we introduce the generalised concept of states (density matrices), observables (POVMs) and transformations (channels), but we also characterise these structures from an algebraic standpoint, which provides many useful technical tools, and clarity as to their generality. This approach also makes it manifest that quantum theory is a direct generalisation of probability theory, and provides a unifying formalism for both fields. The focus on finite-dimensional systems allows for a self-contained presentation which avoids many of the technicalities inherent to the more general $C^*$-algebraic approach, while being appropriate for the quantum information literature.

arXiv:1505.03106May 12, 201510 citations

C^*

-algebraic approach, while being appropriate for the quantum information literature.