- ホーム
- > 洋書
- > 英文書
- > Science / Mathematics
Full Description
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.
Contents
Hilbert space basics
Tensor products of Hilbert spaces
Monotonicity and convexity for operators
von Neumann algebras on finite dimensional Hilbert spaces
Positive linear maps and quantum operators
Some basic trace function inequalities
Fundamental entropy inequalities
Consequences and refinements of SSA
Quantification of entanglement
Convexity, concavity and monotonicity
Majorization methods
Tomita-Takesaki theory and operator inequalities
Convex geometry
Complex interpolation
Bibliography
Index
