Quantum F Divergences In Von Neumann Algebras
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Quantum F Divergences In Von Neumann Algebras
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Author : Fumio Hiai
language : en
Publisher: Springer Nature
Release Date : 2021-01-26
Quantum F Divergences In Von Neumann Algebras written by Fumio Hiai and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-01-26 with Science categories.
Relative entropy has played a significant role in various fields of mathematics and physics as the quantum version of the Kullback–Leibler divergence in classical theory. Many variations of relative entropy have been introduced so far with applications to quantum information and related subjects. Typical examples are three different classes, called the standard, the maximal, and the measured f-divergences, all of which are defined in terms of (operator) convex functions f on (0,∞) and have respective mathematical and information theoretical backgrounds. The α-Rényi relative entropy and its new version called the sandwiched α-Rényi relative entropy have also been useful in recent developments of quantum information. In the first half of this monograph, the different types of quantum f-divergences and the Rényi-type divergences mentioned above in the general von Neumann algebra setting are presented for study. While quantum information has been developing mostly in the finite-dimensional setting, it is widely believed that von Neumann algebras provide the most suitable framework in studying quantum information and related subjects. Thus, the advance of quantum divergences in von Neumann algebras will be beneficial for further development of quantum information. Quantum divergences are functions of two states (or more generally, two positive linear functionals) on a quantum system and measure the difference between the two states. They are often utilized to address such problems as state discrimination, error correction, and reversibility of quantum operations. In the second half of the monograph, the reversibility/sufficiency theory for quantum operations (quantum channels) between von Neumann algebras via quantum f-divergences is explained, thus extending and strengthening Petz' previous work. For the convenience of the reader, an appendix including concise accounts of von Neumann algebras is provided.
An Invitation To Von Neumann Algebras
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Author : V.S. Sunder
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
An Invitation To Von Neumann Algebras written by V.S. Sunder and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Why This Book: The theory of von Neumann algebras has been growing in leaps and bounds in the last 20 years. It has always had strong connections with ergodic theory and mathematical physics. It is now beginning to make contact with other areas such as differential geometry and K-Theory. There seems to be a strong case for putting together a book which (a) introduces a reader to some of the basic theory needed to appreciate the recent advances, without getting bogged down by too much technical detail; (b) makes minimal assumptions on the reader's background; and (c) is small enough in size to not test the stamina and patience of the reader. This book tries to meet these requirements. In any case, it is just what its title proclaims it to be -- an invitation to the exciting world of von Neumann algebras. It is hoped that after perusing this book, the reader might be tempted to fill in the numerous (and technically, capacious) gaps in this exposition, and to delve further into the depths of the theory. For the expert, it suffices to mention here that after some preliminaries, the book commences with the Murray - von Neumann classification of factors, proceeds through the basic modular theory to the III). classification of Connes, and concludes with a discussion of crossed-products, Krieger's ratio set, examples of factors, and Takesaki's duality theorem.
Entropy Divergence And Majorization In Classical And Quantum Thermodynamics
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Author : Takahiro Sagawa
language : en
Publisher: Springer Nature
Release Date : 2022-03-23
Entropy Divergence And Majorization In Classical And Quantum Thermodynamics written by Takahiro Sagawa and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-03-23 with Science categories.
Rich information-theoretic structure in out-of-equilibrium thermodynamics exists in both the classical and quantum regimes, leading to the fruitful interplay among statistical physics, quantum information theory, and mathematical theories such as matrix analysis and asymptotic probability theory. The main purpose of this book is to clarify how information theory works behind thermodynamics and to shed modern light on it. The book focuses on both purely information-theoretic concepts and their physical implications. From the mathematical point of view, rigorous proofs of fundamental properties of entropies, divergences, and majorization are presented in a self-contained manner. From the physics perspective, modern formulations of thermodynamics are discussed, with a focus on stochastic thermodynamics and resource theory of thermodynamics. In particular, resource theory is a recently developed field as a branch of quantum information theory to quantify “useful resources” and has an intrinsic connection to various fundamental ideas of mathematics and information theory. This book serves as a concise introduction to important ingredients of the information-theoretic formulation of thermodynamics.
A Note On The Type Of Local Von Neumann Algebras In Quantum Field Theory Over Curved Space Time
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Author : Manfred Wollenberg
language : de
Publisher:
Release Date : 1992
A Note On The Type Of Local Von Neumann Algebras In Quantum Field Theory Over Curved Space Time written by Manfred Wollenberg and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with categories.
Stone Spectra Of Von Neumann Algebras And Foundation Of Quantum Theory
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Author : Andreas Döring
language : en
Publisher:
Release Date : 2005
Stone Spectra Of Von Neumann Algebras And Foundation Of Quantum Theory written by Andreas Döring and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with categories.
Modular Intersections Of Von Neumann Algebras In Quantum Field Theory
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Author : Hans-Werner Wiesbrock
language : en
Publisher:
Release Date : 1996
Modular Intersections Of Von Neumann Algebras In Quantum Field Theory written by Hans-Werner Wiesbrock and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with categories.
The Kadison Singer Property
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Author : Marco Stevens
language : en
Publisher: Springer
Release Date : 2016-11-07
The Kadison Singer Property written by Marco Stevens and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-11-07 with Science categories.
This book gives a complete classification of all algebras with the Kadison-Singer property, when restricting to separable Hilbert spaces. The Kadison-Singer property deals with the following question: given a Hilbert space H and an abelian unital C*-subalgebra A of B(H), does every pure state on A extend uniquely to a pure state on B(H)? This question has deep connections to fundamental aspects of quantum physics, as is explained in the foreword by Klaas Landsman. The book starts with an accessible introduction to the concept of states and continues with a detailed proof of the classification of maximal Abelian von Neumann algebras, a very explicit construction of the Stone-Cech compactification and an account of the recent proof of the Kadison-Singer problem. At the end accessible appendices provide the necessary background material. This elementary account of the Kadison-Singer conjecture is very well-suited for graduate students interested in operator algebras and states, researchers who are non-specialists of the field, and/or interested in fundamental quantum physics.
Operator And Norm Inequalities And Related Topics
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Author : Richard M. Aron
language : en
Publisher: Springer Nature
Release Date : 2022-08-10
Operator And Norm Inequalities And Related Topics written by Richard M. Aron and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-08-10 with Mathematics categories.
Inequalities play a central role in mathematics with various applications in other disciplines. The main goal of this contributed volume is to present several important matrix, operator, and norm inequalities in a systematic and self-contained fashion. Some powerful methods are used to provide significant mathematical inequalities in functional analysis, operator theory and numerous fields in recent decades. Some chapters are devoted to giving a series of new characterizations of operator monotone functions and some others explore inequalities connected to log-majorization, relative operator entropy, and the Ando-Hiai inequality. Several chapters are focused on Birkhoff–James orthogonality and approximate orthogonality in Banach spaces and operator algebras such as C*-algebras from historical perspectives to current development. A comprehensive account of the boundedness, compactness, and restrictions of Toeplitz operators can be found in the book. Furthermore, an overview of the Bishop-Phelps-Bollobás theorem is provided. The state-of-the-art of Hardy-Littlewood inequalities in sequence spaces is given. The chapters are written in a reader-friendly style and can be read independently. Each chapter contains a rich bibliography. This book is intended for use by both researchers and graduate students of mathematics, physics, and engineering.
C Algebras And W Algebras
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Author : Shôichirô Sakai
language : en
Publisher: Springer
Release Date : 1971
C Algebras And W Algebras written by Shôichirô Sakai and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971 with C*-algebras categories.
From the reviews: "This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert." (Math. Reviews) "In theory, this book can be read by a well-trained third-year graduate student - but the reader had better have a great deal of mathematical sophistication. The specialist in this and allied areas will find the wealth of recent results and new approaches throughout the text especially rewarding." (American Scientist) "The title of this book at once suggests comparison with the two volumes of Dixmier and the fact that one can seriously make this comparison indicates that it is a far more substantial work that others on this subject which have recently appeared"(BLMSoc)
Approximate Quantum Markov Chains
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Author : David Sutter
language : en
Publisher: Springer
Release Date : 2018-04-20
Approximate Quantum Markov Chains written by David Sutter and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-04-20 with Science categories.
This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen the data-processing inequality such that it reveals a statement about the reconstruction of lost information. The main difficulty in order to understand the behavior of quantum Markov chains arises from the fact that quantum mechanical operators do not commute in general. As a result we start by explaining two techniques of how to deal with non-commuting matrices: the spectral pinching method and complex interpolation theory. Once the reader is familiar with these techniques a novel inequality is presented that extends the celebrated Golden-Thompson inequality to arbitrarily many matrices. This inequality is the key ingredient in understanding approximate quantum Markov chains and it answers a question from matrix analysis that was open since 1973, i.e., if Lieb's triple matrix inequality can be extended to more than three matrices. Finally, we carefully discuss the properties of approximate quantum Markov chains and their implications. The book is aimed to graduate students who want to learn about approximate quantum Markov chains as well as more experienced scientists who want to enter this field. Mathematical majority is necessary, but no prior knowledge of quantum mechanics is required.