Completeness of inner product spaces and quantum logic of splitting subspaces

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which EE =S holds, is a quantum logic, that is, an orthocomplemented orthomodular -orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys. 11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces.     

The free energy of systems with net charge

For quite general domains, the free energy density of a system with net charge differs in the thermodynamic limit from that of a neutral system by an electrostatic correction.     

The dynamical entropy of the quantum Arnold cat map

We present a rigorous computation of the dynamical entropyh of the quantum Arnold cat map. This map, which describes a flow on the noncommutative two-dimensional torus, is a simple example of a quantum dynamical system with optimal mixing properties, characterized by Lyapunov exponents ± 1n ⁺, ⁺ > 1. We show that, for all values of the quantum deformation parameter,h coincides with the positive Lyapunov exponent of the dynamics.     

Defining quantum dynamical entropy

We propose an elementary definition of the dynamical entropy for a discrete-time quantum dynamical system. We apply our construction to classical dynamical systems and to the shift on a quantum spin chain. In the first case, we recover the Kolmogorov-Sinai invariant and, for the second, we find the mean entropy of the invariant state plus the logarithm of the dimension of the single-spin space.     

A Phase Transition in a Quantum Crystal with Asymmetric Potentials

A translation invariant system of interacting quantum anharmonic oscillators indexed by the elements of a simple cubic lattice is considered. The anharmonic potential is of general type, which in particular means that it might have no symmetry. For this system, we prove that the global polarization (obtained in the thermodynamic limit) gets discontinuous at a certain value of the external field provided d ≥ 3, and the particle mass as well as the interaction intensity are big enough. The proof is based on the representation of local Gibbs states in terms of path measures and thereby on the use of the infrared estimates and the Garsia–Rodemich–Rumsey inequality.      

On the Riemannian metric of α-entropies of density matrices

The Riemannian metric induced by quantum -entropies is proven to be monotone under stochastic mappings on the set of density matrices. The length of tangent vectors is essentially the Wigner-Yanase-Dyson skew information in this setting.Supported by the Hungarian National Foundation for Scientific Research, grant No. T-016924.     

Abundance of translation invariant pure states on quantum spin chains

We construct a set of translation invariant pure states of a quantum spin chain, which is w -dense in the set of all translation invariant states of the chain. Each of the approximating states has exponential decay of correlations, and is the unique ground state of a finite range Hamiltonian with a spectral gap above the ground state energy.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Finite energy sectors of the XY model

Continuing the earlier work on soliton sectors, we determine all finite energy representations of the XY model for almost all parameter values. In the interior of unique vacuum regions of parameters (i.e. the large external magnetic field region ||>1), the unique irreducible vacuum representation is the only finite energy representation.At the critical values of the parameters (||=1 as well as theXY symmetric case =0, ||1), there is an infinite number of mutually nonequivalent irreducible finite energy representations. Apart from the unique irreducible ground state representation and another associated irreducible representation, these infinite number of representations arise from an infinite number of nearly zero energy excitations of the ground state with a finite total energy and may be called infrared representations.In the remaining cases, as have been studied earlier, there are two additional irreducible finite energy representations besides two irreducible ground state representations and they are topological soliton sectors with different ground state limits in positive and negative spatial infinity. (For two exceptional values of parameters (, )=(0, ±1), they also become ground state representations.)     

Deterministic quantum noise and Kolmogorov systems

Two irreversible quantum evolutions with zero dynamical entropy, when dilated to reversible automorphisms, provide quantum Kolmogorov systems of both the algebraic and entropic type with infinite-dynamical entropy.     

An example of an infinite system with zero quantum dynamical entropy

With nary mention of a tree graph, we obtain a cluster expansion bound that includes and vastly generalizes bounds as obtained by extant tree graph inequalities. This includes applications to both two-body and many-body potential situations of the recently obtained new improved tree graph inequalities that have led to the extra 1/N! factors. We work in a formalism coupling a discrete set of boson variables, such as occurs in a lattice system in classical statistical mechanics, or in Euclidean quantum field theory. The estimates of this Letter apply to numerical factors as arising in cluster expansions, due to essentially arbitrary sequences of the basic operations: interpolation of the covariance, interpolation of the interaction, and integration by parts. This includes complicated evolutions, such as the repeated use of interpolation to decouple the same variables several times, to ensure higher connectivity for renormalization purposes, in quantum field theory.This work was supported in part by the National Science Foundation under grant no. PHY-87-01329.     

Maximizing relative entropy at given energies

We consider maximization of the relative entropy (with respect to a fixed normal state) in a von Neumann algebra among the states having fixed expectation for finitely many self-adjoint elements.     

The dynamics of infinite open quantum systems

We establish the existence of the thermodynamic limit of quantum dynamical semigroups describing the irreversible dynamics of a class of mean-field quantum systems coupled to collective or individual reservoirs.     

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher–Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.     

Semi-indefinite inner product and generalized Minkowski spaces

In this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.     

New boson representations of the sl q (2) with multiplicity two and new solutions to the Yang-Baxter equation atq p =1

Whenq is a root of unity, the representations of the quantum universal enveloping algebra sl _q (2) with multiplicity two are constructed from theq-deformed boson realization with an arbitrary parameter which is in a very general form and is first presented in this Letter. The new solutions to the Yang-Baxter equation are obtained from these representations through the universalR-matrix.This work is supported in part by the National Foundation of Natural Science of China.     

Non-zero Entropy Density in the <Emphasis Type="Italic">XY</Emphasis> Chain Out of Equilibrium

The von Neumann entropy density of a block of n spins is proved to be non-zero for large n in the non-equilibrium steady state of the XY chain constructed by coupling a finite cutout of the chain to the two infinite parts to its left and right which act as thermal reservoirs at different temperatures. Moreover, the non-equilibrium density is shown to be strictly greater than the density in thermal equilibrium.      

The critical temperature and gap solution in the Bardeen-Cooper-Schrieffer theory of superconductivity

The Letter studies the problem of numerical approximations of the critical transition temperature and the energy gap function in the Bardeen-Cooper-Schrieffer equation arising in superconductivity theory. The positive kernel function leads to a phonon-dominant state at zero temperature. Much attention is paid to the equation defined on a bounded region. Two discretized versions of the equation are introduced. The first version approximates the desired solution from below, while the second, from above. Numerical examples are presented to illustrate the efficiency of the method. Besides, the approximations of a full space solution and the associated critical temperature by solution sequences constructed on bounded domains are also investigated.Part of this work was done while this author was visiting the Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA