On the mixing-enhancing dynamical maps and mixing-enhancing evolution of a quantum system

The mixing-enhancing (in the sense of Uhlmann) dynamical maps and dynamical evolution is studied. We give a necessary and sufficient condition for a dynamical map (and dynamical evolution) of a quantum system to be mixing-enhancing. In the case of a finite- dimensional Hilbert space this condition is equivalent to the condition that the dynamical map (dynamical evolution) preserve the most mixed state and the von Neumann entropy be non- decreasing. It is proved that, in contrast with the finite-dimensional case, increasing of the von Neumann entropy under a dynamical map (for any initial state) does not imply that the dynamical map is mixing-enhancing. We also give a necessary and sufficient condition for an infinitesimal generator of a norm-continuous dynamical semigroup to be mixing-enhancing.     

Renormalization and asymptotic behavior in a finite quantum field theory with shadow states

In this paper we present a study of the renormalization problem in a finite quantum field theory with shadow states for a system of a physical scalar field interacting with a physical fermion field. In order to make the theory finite, two fermion shadow fields are introduced. We observe that the stability criterion of renormalization can not be satisfied simultaneously by both physical fields and shadow fields, if the finiteness of the theory is to be maintained. A physical interpretation of this result is given. Furthermore, we find that the effective complete propagators for large space-like momenta behave like free field propagators without the logarithmic factors observed in the non-abelian gauge theory.     

On the asymptotic behavior of effective QED at finite temperature

The aim of this paper is to study finite temperature effects in effective quantum electrodynamics using Weisskopfs zero-point energy method in the context of thermo field dynamics. After a general calculation for a weak magnetic field at fixed T, the asymptotic behavior of the Euler-Kockel-Heisenberg Lagrangian density is investigated focusing on the regularization requirements in the high temperature limit. In scalar QED the same problem is also discussed.Received: 14 June 2003, Revised: 14 August 2003, Published online: 2 October 2003This revised version was published online in Oktober 2003 with corrections to the citation of equations in the text after equation (17).     

Quantum families of maps and quantum semigroups on finite quantum spaces

Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.     

On the generators of quantum dynamical semigroups

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() is derived. This is a quantum analogue of the Lévy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

On the asymptotic behavior in the scattering problem for several charged quantum particles interacting via repulsive pair potentials

The ansatz proposed earlier by Buslaev and Levin [Funct. Anal. Appl. 46, 147 (2012)] for describing the leading order in the asymptotic behavior of continuum eigenfunctions in the scattering problem for three charged quantum particles is extended to the system of n charged quantum particles of identical mass and identical charge. It is shown that the proposed ansatz generates a fast decreasing (faster than the potential) discrepancy of the Schrödinger equation for a set of asymptotic configurations.     

On the possible temperatures of a dynamical system

A simpleC*-algebra and a continuous one-parameter automorphism group are constructed such that the set of inverse temperatures at which there exist equilibrium states (i.e., KMS states, or, for =±, ground or ceiling states) is an arbitrary closed subset of IR{±}.With partial support of the National Science Foundation     

On positive maps in quantum information

Using the Grothendieck approach to the tensor product of locally convex spaces, we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented.     

On the asymptotic behavior of an earthquake model

We present evidence showing that there are important differences in the relaxation times associated with different aspects of the model of Olami, Feder and Christensen. The asymptotic behavior is characterized by a critical exponent that is independent of α, the degree of conservation of the model. The analysis of temporal series of avalanches gives helpful information to study the approach to self-organized criticality, and to put in evidence any quasi-periodic evolution.     

On one asymptotic property of time-shift quantum cryptography

One of the basic results of classical information theory is that error-free information transmission is possible even through an imperfect binary communication channel with noise up to an error of Q _c = 1/2. There is a fundamental and applied question of whether quantum-mechanical constraints can ensure error-free classical-information transmission with quantum states and, moreover, guarantee the security of distributed keys up to the theoretical limit in the error Q _c. It has been shown that the secure key distribution is possible up to the error Q _c in the asymptotic limit of a large number of bases.     

On asymptotic properties of a quantum number ω in a system with SU(3) symmetry

The asymptotic properties of the basic vectors for the (λμ) irreducible representations of the SU(3)-group in the SU(3)?SO(3) reduction (Elliott's basis) are investigated for large values of λ or μ. The matrix of the Bargmann-Moshinsky operator Ω in Elliott's basis is analysed for the same limiting case. Approximate expressions for the eigenvalues and eigenvectors of the operator Ω are found. They demonstrate the asymptotic properties of the operator Ω to be very close to those of the rigid quantum top.     

Semi-classical approximation for a class of quantum dynamical semigroups

We determine the complex semi-classical approximation to the propagator for a class of quantum dynamical semigroups and study a simple model. The method uses functional integral techniques developed earlier.     

On the asymptotic behavior of the pair correlation function for liquid4He

A new expression for the asymptotic pair correlation function of He II is proposed. Unlike the standard expressions, this contains explicitly the condensate fraction and leans heavily on the corresponding function for He I. The consequences for the long-wavelength liquid structure factor and the asymptotic effective interaction in the system are explored. In passing, the triplet and higher-order correlations are discussed briefly. Whenever possible, the connection with other treatments is pointed out.     

On the asymptotic behavior of the solutions of the Caldirola-Kanai equation

We study in this Letter the asymptotic behavior, as t+, of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition V(x)+ as |x|. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability P _t (t) of finding the particle inside I tends to zero as t+. On the other hand, when I contains critical points of V in its interior, we prove that P _t (t) does not oscillate indefinitely, but tends to a limit as t+. In particular, when the potential has only isolated critical points x ₁, ..., x _N our results imply that the probability density of the particle tends to in the sense of distributions.Supported by Fulbright-MEC grant 85-07391.