Random Operators and Crossed Products

This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review several formulas for this trace, show how it comes as an application of Connes" noncommutative integration theory and discuss Shubin"s trace formula. We then restrict ourselves to the case of an action of a group on a group and include new proofs for some theorems of Bellissard and Testard on an analogue of the classical Plancherel theorem. We show that the integrated density of states is a spectral measure in the periodic case, thereby generalizing a result of Kaminker and Xia. Finally, we discuss duality results and apply a method of Gordon et al. to establish a duality result for crossed products by Z.     

Continuity of the von Neumann Entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.     

The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr?dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schr?dinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schr?dinger operators with Neumann boundary conditions. Received: 20 October 2000 / Accepted: 8 March 2001     

On the independence of local algebras II

The connection between independence of von Neumann algebras and their commutation is studied, without particular assumptions on representations of these algebras. The condition equivalent to the commutation of von Neumann algebras of operators in Hilbert space, formulated in language of operations over states, is given.     

Uncertainty Relations for Coherence

Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases(MUBs), and general symmetric informationally complete positive operator valued measurements(SIC-POVMs),respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance,which are of hybrid nature(mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement(reference basis) such that the coherence of these states can be simultaneously small.     

States on projection logics of von Neumann algebras

We summarize recent results concerning states on projection lattices of von Neumann algebras. In particular, we present an analysis of the Jauch-Piron property in the von Neumann algebra setting.     

Quantitative conditions for time evolution in terms of the von Neumann equation

The adiabatic theorem describes the time evolution of the pure state and gives an adiabatic approximate solution to the Schr ¨odinger equation by choosing a single eigenstate of the Hamiltonian as the initial state. In quantum systems, states are divided into pure states(unite vectors) and mixed states(density matrices, i.e., positive operators with trace one). Accordingly, mixed states have their own corresponding time evolution, which is described by the von Neumann equation. In this paper, we discuss the quantitative conditions for the time evolution of mixed states in terms of the von Neumann equation. First, we introduce the definitions for uniformly slowly evolving and δ-uniformly slowly evolving with respect to mixed states, then we present a necessary and sufficient condition for the Hamiltonian of the system to be uniformly slowly evolving and we obtain some upper bounds for the adiabatic approximate error. Lastly, we illustrate our results in an example.     

A NEW MEASURE FOR THE JAYNES-CUMMINGS MODEL DYNAMICS DISTANCE BETWEEN DENSITY OPERATORS

We use the distance between density operators to study the dynamical evolution of the Jaynes-Cummings model with an additional Kerr medium, and to compare the result with the corresponding von Neumann entropy. We have shown that the distance between density operators can provide more detailed information about the dynamical behavior of the quantum system than von Neumann entropy.     

A NEW MEASURE FOR THE JAYNES-CUMMINGS MODEL DYNAMICS --- DISTANCE BETWEEN DENSITY OPERATORS

We use the distance between density operators to study the dynamical evolution of the Jaynes-Cummings model with an additional Kerr medium, and to compare the result with the corresponding von Neumann entropy. We have shown that the distance between density operators can provide more detailed information about the dynamical behavior of the quantum system than von Neumann entropy.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

Existence of the Density of States for Multi-Dimensional¶Continuum Schrödinger Operators with¶Gaussian Random Potentials

A Wegner estimate is proved for quantum systems in multi-dimensional Euclidean space which are characterized by one-particle Schr?dinger operators with random potentials that admit a certain one-parameter decomposition. In particular, the Wegner estimate applies to systems with rather general Gaussian random potentials. As a consequence, these systems possess an absolutely continuous integrated density of states, whose derivative, the density of states, is locally bounded. An explicit upper bound is derived. Received: 13 November 1996 / Accepted: 30 April 1997     

Wegner Estimate and the Density of States of Some Indefinite Alloy-Type Schrödinger Operators

We study Schrödinger operators with a random potential of alloy type. The single site potentials are allowed to change sign. For a certain class of them, we prove a Wegner estimate. This is a key ingredient in an existence proof of pure point spectrum of the considered random Schrödinger operators. Our estimate is valid for all bounded energy intervals and all space dimensions and implies the existence of the density of states.     

Lifshitz Tails for Acoustic Waves in Random Quantum Waveguide

In this study, we consider acoustic operators in a random quantum waveguide. Precisely we deal with an elliptic operator in the divergence form on a random strip. We prove that the integrated density of states of the relevant operator exhibits Lifshitz behavior at the bottom of the spectrum. This result could be used to prove localization of acoustic waves at the bottom of the spectrum. 2000 Mathematics Subject Classification: 81Q10, 35P05, 37A30, 47F05     

Algebras of bounded operators on nonclassical orthomodular spaces

A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite dimensional examples constructed over certain non-Archimedeanly valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebraA of von Neumann type that contains no orthogonal projections at all. For operators inA we establish a representation theorem from which we deduce thatA is commutative. We then focus on a subalgebra which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of are given.     

Different Types of Conditional Expectation and the Lüders—von Neumann Quantum Measurement

In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lüders—von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lüders—von Neumann measurement are presented, and the question whether they coincide is studied. All this is done in the general framework of Jordan operator algebras. The examples considered include the type I and type II operator algebras, the standard Hilbert space model of quantum mechanics, and a no-go result concerning the conditional expectation of observables that satisfy the canonical commutator relation.     

Operatorial form of the theory of polarization optical devices: I. Spectral theory of the basic devices

While the theory of operators in quantum mechanics is expressed nowadays in a pure operatorial form (wrapped mostly in Dirac's symbolic language), in optics the polarization device operators and their action are analyzed yet in the old matrix (Jones or Muller) formalism. The theory of polarization device operators has not taken systematically advantage of the very general, fundamental and deep results of the spectral theory of operators, on the basis of which it can be structured in an elegant deductive and physically expressive form. In this paper we apply the spectral theorem to the polarization device operators, we calculate their expansions in a pure operatorial Dirac-dyadic form and give some examples which illustrate the advantages from the physical insight viewpoint of such an approach. We are concerning here only with the basic polarization devices, to which correspond normal operators.     

Dynamical entanglement for Fermi coupled stretching and bending modes

The dynamical entanglement for Fermi coupled C--H stretch and bend vibrations in molecule CHD₃ is studied in terms of two negativities and the reduced von Neumann entropy, where initial states are taken to be direct products of photon-added coherent states on each mode. It is demonstrated that the negativity defined by the sum of negative eigenvalues of the partial transpose of density matrices is positively correlated with the von Neumann entropy. The entanglement difference between photon-added coherent states and usual coherent states is discussed as well.     

Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.     

Quantification of correlations in quantum many-particle systems

We introduce a well-defined and unbiased measure of the strength of correlations in quantum many-particle systems which is based on the relative von Neumann entropy computed from the density operator of correlated and uncorrelated states. The usefulness of this general concept is demonstrated by quantifying correlations of interacting electrons in the Hubbard model and in a series of transition-metal oxides using dynamical mean-field theory.