Inequalities in Matrix Algebras

Eric Carlen

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English

American Mathematical Society
30 September 2025

Algebra; Geometry

Series: Graduate Studies in Mathematics

The theory of positive or completely positive maps from one matrix algebra to another is the mathematical theory underlying the quantum mechanics of finite systems, as well as much of quantum information and computing. Inequalities are fundamental to the subject, and a watershed event in its development was the proof of the strong subadditivity of quantum entropy by Lieb and Ruskai. Over the next 50 years, this result has been extended and refined extensively. The development of the mathematical theory accelerated in the 1990s when researchers began to intensively investigate the quantum mechanical notion of """"entanglement"""" of vectors in tensor products of Hilbert spaces. Entanglement was identified by Schrodinger as a fundamental aspect of quantum mechanics, and in recent decades questions about entanglement have led to much mathematical progress. What has emerged is a beautiful mathematical theory that has very recently arrived at a mature form.

This book is an introduction to that mathematical theory, starting from modest prerequisites. A good knowledge of linear algebra and the basics of analysis and probability are sufficient. In particular, the fundamental aspects of quantum mechanics that are essential for understanding how a number of questions arose are explained from the beginning.

By: Eric Carlen
Imprint: American Mathematical Society
Country of Publication: United States [Currently unable to ship to USA: see Shipping Info]
Dimensions: Height: 254mm, Width: 178mm,
ISBN: 9781470480264
ISBN 10: 1470480263
Series: Graduate Studies in Mathematics
Pages: 458
Publication Date: 30 September 2025
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Forthcoming

Eric Carlen, Rutgers University, New Brunswick, NJ.