The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Levy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.

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On the Generators of Quantum Dynamical Semigroups

In information processing, as in physics, our classical world view provides an incomplete approximation to an underlying quantum reality. Quantum effects like interference and entanglement play no direct role in conventional information processing, but they can--in principle now, but probably eventually in practice--be harnessed to break codes, create unbreakable codes, and speed up otherwise intractable computations.

Quantum information and computation

Index for subfactors

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Positive Definite Matrices

We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Neel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

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Finitely correlated states on quantum spin chains

Comparison theory of projections--exercises and solutions Normal states and unitary equivalence of von Neumann algebras--exercises and solutions The trace--exercises and solutions Algebra and commutant--exercises and solutions Special representations of $C^*$-algebras--exercises and solutions Tensor products--exercises and solutions Approximation by matrix algebras--exercises and solutions Crossed products--exercises and solutions Direct integrals and decomposiitons--exercises and solutions Bibliography Index.

Fundamentals of the Theory of Operator Algebras

In [2]2 we classified the isometric mappings of one C*-algebra (uniformly closed, self-adjoint operator algebra) onto another. It was remarked in that paper that the results obtained were a non-commutative extension of results of Banach [1] and Stone [7]. While this was true in spirit, we were well aware that it was not accurate to the letter. Banach and Stone deal with the algebra of real continuous functions on a compact-Hausdorff space, and our results concerning C*-algebras are actually the non-commutative analogue of results concerning the complex function algebra. The strict non-commutative analogue of the real function algebra is the Jordan algebra of self-adjoint elements in a C*-algebra (Jordan C*-algebra). The complex and real theorems follow very easily from one another in the commutative case, so that one might justifiably consider the C*-algebra theorem an extension of both of the function algebra theorems. Despite such trifling considerations, two questions still remain: what are the isometries of one C*-algebra onto another, and what are the isometries of one Jordan C*-algebra onto another? At the time [21 was written, the C*-algebra seemed the more natural object to consider. In view of the results obtained, answering the Jordan C*-algebra questions appeared to be an unnecessary decoration to the theory. We felt that the Jordan C*-algebra results could be obtained from the C*-algebra results in the same way that the real function algebra theorem follows from the complex function algebra theorem (viz., by showing that the complexified linear map is everywhere isometric). Subsequent investigations have changed our attitude in this matter. An important application of these considerations requires a Jordan C*-algebra theorem for one thing, and our attempts to derive this theorem directly from the C*-algebra theorem failed for another. The result in question is contained in Theorem 2 of ?2 and states (in normalized form) that an isometry between two Jordan C*-algebras which carries the identity into the identity is a C* (Jordan) -isomorphism. This theorem was eventually proved with the aid of a Generalized Schwarz Inequality (cf. Theorem 1 of ?2). In effect, an alternative ending has been given to the proof of [Theorem 7; 2]. This ending is by no means simpler or shorter than the one given in [2] (though it is, perhaps, less contrived), but it is flexible enough to allow us to draw the desired Jordan C*-algebra conclusion. The critical application of these results is contained in Corollary 3. A discussion accompanies Corollaries 3 and 4, but a few additional remarks are in order.

Richard V. Kadison

Papers

Fundamentals of the Theory of Operator Algebras

A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras

A representation theory for commutative topological algebra

Extensions of pure states.

Fundamentals of the Theory of Operator Algebras. Volume IV