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By M. Ohya, Denes Petz
This monograph, by way of now a typical reference within the box, offers an entire evaluation of quantum entropy thought, from the early days of quantum mechanics as much as the newest advancements. It employs mathematical formalism according to operator conception and practical research with a objective of constructing a finished account of the several entropy strategies, their interrelations and use. one of the rules mentioned are entropy of a quantum nation, relative entropy, mutual entropy, entropy of a rough graining and dynamical entropy. quite a few basic homes of quantum info size are built, together with the von Neumann entropy of a statistical operator and its restricting normalized model, the entropy expense. Use of quantum-entropy amounts is made in perturbation conception, vital restrict theorems, thermodynamics of spin platforms, entropic uncertainty family, and optical communique. This new softcover corrected reprint includes summaries of modern advancements additional to the ends of the chapters.
Content point » Research
Keywords » Entropie - Entropy
Related matters » utilized & Technical Physics - Quantum Physics - Theoretical and Computational Chemistry
Cover
Quantum Entropy and Its Use
Copyright
© Springer-Verlag Berlin Heidelberg 1993
ISBN 3-540-54881-5
ISBN 0-387-54881-5
Preface
Contents
Introduction
Notes and Remarks
Part I: Entropies for Finite Quantum Systems
1. primary Concepts
Notes and Remarks
2. Postulates for Entropy and Relative Entropy
Notes and Remarks
three. Convex hint Functions
Notes and Remarks.
Part II: Entropies for normal Quantum Systems
four. Modular thought and Auxiliaries
Notes and Remarks
five. Relative Entropy of States of Operator Algebras
Notes and Remarks.
6. From Relative Entropy to Entropy
Notes and Remarks
7. Functionals of Entropy Type
Notes and Remarks
Part III: Channeling Transformation and Coarse Graining
eight. Channels and Their Transpose
Notes and Remarks
nine. adequate Channels and Measurements
Notes and Remarks
10. Dynamical Entropy
Notes and Remarks
eleven. desk bound Processes
Notes and Remarks.
Part IV: Perturbation Theory
12. Perturbation of States
Notes and Remarks
thirteen. Variational Expression of Perturbational Limits
Notes and Remarks.
Part V: Miscellanea
14. imperative restrict and Quasifree Reduction
Notes and Remarks.
15. Thermodynamics of Quantum Spin Systems
Notes and Remarks.
sixteen. Entropic Uncertainty Relations
Notes and Remarks.
17. Temperley-Lieb Algebras and Index
Notes and Remarks.
18. Optical verbal exchange Processes
Notes and Remarks.
Bibliography
Subject Index
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Extra resources for Quantum Entropy and Its Use
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10. If F"','PI\ — = for 'P = 'Pc > otherwise, then log Tn where 'Pc is the canonical state given by the density exp(—,3(H — The proof follows from the observation Proof. 1), we have and equality holds only in the case 'P = 0 'Pc. 10 to relative entropy we introduce perturbed functionals. Let 'P be a state (or a positive functional) on a finite quantum system and let h e A be selfadjoint. Set for the density and p for the support projection of 'Ph is given by the density matrix p exp(log + php) is understood as where log not normalized and supp 'Ph = supp h= The the perturbed functional 'P + p')).
Then the functional a fixed observ- FL(A) = Tr exp(L + log A) A defined on Proof. is concave. The function f(x) = Tr exp(L + xK) of the real variable x is differentiable in a neighbourhod of 0. The theorem is equivalent to the statement d2 L dx2 x=O <0 for all choices of S E Indeed, if f is concave then putting and K, L E A=A2 and K=A1 —A2 weget Tr exp(L + log(A2 + A(A1 — A2)) ATr exp(L + log(A2 + (A1 — A2))) + (1 — A)Tr exp(L + log A2). To compute the second derivative of f we may benefit from the norm convergent expansions log(A + xK) = logA + XTA(K) — + and = ea +I I JoJo +I Jo With the simplification B +...
Instead of selecting a subsequence we suppose that Q(n(k)) T weakly. 27) the weak limit Q(n(k))P(n(k), 6) —* T follows. Consequently, lim k—*oo P(n(k), 6)) w(T) 1 — e. 30) 1. 30) and we arrive at 0 a contradiction with (ii) if 0 <6