Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.

Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.

Differential (and more general self-adjoint) operators involving singular interactions arise naturally in a range of topics such as, classical and quantum physics, chemistry, and electronics. This book presents a systematic mathematical study of these operators, with particular emphasis on spectral and scattering problems. Suitable for researchers in analysis or mathematical physics, this book could also be used as a text for an advanced course on the applications of analysis.

In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space 1 X over a Riemannian manifold X. This geometry is``non-flat'' even if X=R d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient { 1 , divergence div 1 , and Laplace Beltrami operator H 1 =&div 1 { 1 are constructed. The associated volume elements, i.e., all measures + on 1 X w.r.t. which { 1 and div 1 become dual operators on L 2 (1 X ; +), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on 1 X . The corresponding Dirichlet forms E 1 + on L 2 (1 X ; +) are, therefore, defined. Each is shown to be associated with a diffusion process which is thus the Brownian motion on 1 X and which is subsequently identified as the usual independent infinite particle process on X. The associated heat semigroup (T 1 + (t)) t>0 is calculated explicitly. It is also proved that the diffusion process, when started with +, is time-ergodic (or equivalently E 1 + is irreducible or equivalently (T 1 + (t)) t>0 is ergodic) if and only if + is Poisson measure ? z dx with intensity z dx for some z 0. Furthermore, it is shown that the Laplace Beltrami operator H 1 =&div 1 { 1 on L 2 (1 X ; ? z dx ) is unitary equivalent to the second quantization of the Laplacian &2 X on X on the corresponding Fock space n 0 L 2 (X; z dx) n . As another direct consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields on X on Poisson space. Finally, generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on X are described, in particular, the case where the mixed Poisson measures + are replaced by Gibbs measures of Ruelle-type on 1 X .

We give an explicit tight lower bound for the entanglement of formation for arbitrary bipartite mixed states by using the convex hull construction of a certain function. This is achieved by revealing a novel connection among the entanglement of formation, the well-known Peres-Horodecki and realignment criteria. The bound gives a quite simple and efficiently computable way to evaluate quantitatively the degree of entanglement for any bipartite quantum state.PACS numbers: 03.67. Mn, 03.65.Ud, 89.70.+c Quantum entangled states are used as key resources in quantum information processing and communication, such as in quantum cryptography, quantum teleportation, dense coding, error correction and quantum computation [1]. A fundamental problem in quantum information theory is how to quantify the degree of entanglement in a practical and operational way [2,3]. One of the most meaningful and physically motivated measures is the entanglement of formation (EOF) [2,3], which quantifies the minimal cost needed to prepare a certain quantum state in terms of EPR pairs. Related to the EOF the behavior of entanglement has recently been shown to play important roles in quantum phase transition for various interacting quantum many-body systems [4] and may significantly affect macroscopic properties of solids [5]. Moreover, it has been shown that there is a remarkable connection between entanglement and the capacity of quantum channels [6]. A quantitative evaluation of EOF is thus of great significance both theoretically and experimentally.Considerable efforts have been spent on deriving EOF or its lower bound through analytical and numerical approaches, for some limited sets of mixed states [7,8,9,10,11,12,13,14,15,16,17]. Among them, the most noteworthy results are an elegant analytical formula for two qubits [7,8] [18,19] in giving analytic lower bounds [that can be optimized further numerically [18]] for the concurrence, which permits to furnish a lower bound of EOF for a generic mixed state. However, this lower bound is not explicit except for the case of 2 ⊗ n systems [19]. For low dimensional systems, numerical methods [10] can be used to estimate EOF. Nevertheless, they are generally time-consuming and often not very efficient. The notorious difficulty of evaluation for EOF is due to the complexity to solve a high dimensional optimization problem, which becomes a formidable task, as the dimensionality of the Hilbert space grows.In this Letter, we present the first analytical calculation of a tight lower bound of EOF for arbitrary bipartite quantum states. An explicit expression for the bound is obtained from the convex hull of a simple function, based on a known result in [9]. This is achieved by establishing a key connection between EOF and two strong separability criteria, the Peres-Horodecki criterion [20,21] and the realignment criterion [22,23]. The bound is shown to be exact for some special states such as isotropic states [9,24] and permits to provide EOF estimations for many bound entangled states (BES). It provi...