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.     

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.     

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.     

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.     

Quantum phase transition and von Neumann entropy of quasiperiodic Hubbard chains

The half-filled Hubbard chains with the Fibonacci and Harper modulating site potentials are studied in a selfconsistent mean-field approximation. A new order parameter is introduced to describe a charge density order. We also calculate the von Neumann entropy of the ground state. The results show that the von Neumann entropy can identify a CDW/SDW （charge density wave/spin density wave） transition for quasiperiodic models.     

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.     

Superderivations and symmetric Markov semigroups

Unbounded superderivations are used to construct non-commutative elliptic operators on semi-finite von Neumann algebras. The method exploits the interplay between dynamical semigroups and Dirichlet forms. The elliptic operators may be viewed as generators of irreversible dynamics for fermion systems with infinite degrees of freedom.     

Dynamic properties of wehrl information entropy and wehrl phase distribution for a moving four-level atom

We study the dynamics of the von Neumann entropy, Wehrl entropy, and Wehrl phase distribution for a single four-level ladder-type atom interacting with a one-mode cavity field taking into account the atomic motion. We obtain the exact solution of the model using the Schr¨odinger equation under specific initial conditions. Also we investigate the quantum and classical quantifiers of this system in the nonresonant case. We examine the effects of detuning and the atomic motion parameter on the entropies and their density operators. We observe an interesting monotonic relation between the different physical quantities in the case of nonmoving and moving atoms during the time evolution. We show that both the detuning and the atomic motion play important roles in the evolution of the Wehrl entropy, its marginal distributions, entanglement, and atomic populations.     

Groupoids,von Neumann Algebras and the Integrated Density of States

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schrödinger operators on manifolds, quantum percolation and quasi–crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure.     

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.     

Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy,Relative Entropy Distance and Energy Constraints

We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: first, a tight analysis of the Alicki–Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications, we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, E_R, and its regularization \({E_R^{\infty}}\), as well as of the entanglement of formation, E_F. Using a novel “quantum coupling” of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, \({E_C=E_F^{\infty}}\). Second, we derive analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.     

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.     

Discretization of the Density Matrix as a Nonlinear Positive Map and Entanglement

We propose the discretization of the density matrix as a nonlinear positive map for systems with continuous variables. We use this procedure for calculating the entanglement between two modes through different criteria, such as Tsallis entropy, von Neumann entropy and linear entropy, and the logarithmic negativity. As an example, we study the dynamics of entanglement for the two-mode squeezed-vacuum state in the parametric amplifier and show good agreement with the analytic results. Also we address the loss of information on the system state due to the discretization of the density matrix.     

Algebras of Random Operators Associated to Delone Dynamical Systems

We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C ^*-subalgebra to discuss a Shubin trace formula.     

Dynamical origin of the quantum Zeno effect

The quantum Zeno effect is often studied and understood in term of nonunitary evolutions, involving projections à la von Neumann (measurements). We propose a dynamical explanation of this effect, which involves only unitary operators. The limit of infinitely frequent measurements is critically discussed: it is unphysical, yet interesting and peculiar.     

Standard quantum mechanics featuring probabilities instead of wave functions

A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.     

Von Neumann entropy in a Rashba-Dresselhaus nanodot; dynamical electronic spin-orbit entanglement

The main purpose of the present article is to report the characteristics of von Neumann entropy, thereby, the electronic hybrid entanglement, in the heterojunction of two semiconductors, with due attention to the Rashba and Dresselhaus spin-orbit interactions. To this end, we cast the von Neumann entropy in terms of spin polarization and compute its time evolution; with a vast span of applications. It is assumed that gate potentials are applied to the heterojunction, providing a two dimensional parabolic confining potential (forming an isotropic nanodot at the junction), as well as means of controlling the spin-orbit couplings. The spin degeneracy is also removed, even at electronic zero momentum, by the presence of an external magnetic field which, in turn, leads to the appearance of Landau states. We then proceed by computing the time evolution of the corresponding von Neumann entropy from a separable (spin-polarized) initial state. The von Neumann entropy, as we show, indicates that electronic hybrid entanglement does occur between spin and two-dimensional Landau levels. Our results also show that von Neumann entropy, as well as the degree of spin-orbit entanglement, periodically collapses and revives. The characteristics of such behavior; period, amplitude, etc., are shown to be determined from the controllable external agents. Moreover, it is demonstrated that the phenomenon of collapse-revivals’ in the behavior of von Neumann entropy, equivalently, electronic hybrid entanglement, is accompanied by plateaus (of great importance in quantum computation schemes) whose durations are, again, controlled by the external elements. Along these lines, we also make a comparison between effects of the two spin-orbit couplings on the entanglement (von Neumann entropy) characteristics. The finer details of the electronic hybrid entanglement, which may be easily verified through spin polarization measurements, are also accreted and discussed. The novel results of the present article, with potent applications in the field of quantum information processing, provide a deeper understanding of the electronic von Neumann entropy and hybrid entanglement that occurs in two-dimensional nanodots.     

Calculation of von Neumann entropy for hydrogen and positronium negative ions

In the present work, we carry out calculations of von Neumann entropies and linear entropies for the hydrogen negative ion and the positronium negative ion. We concentrate on the spatial (electron–electron orbital) entanglement in these ions by using highly correlated Hylleraas functions to represent their ground states, and to take care of correlation effects. We apply the Schmidt decomposition method on the partial-wave expanded two-electron wave functions, and from which the one-particle reduced density matrix can be obtained, leading to the quantifications of linear entropy and von Neumann entropy in the H⁻ and Ps⁻ ions.     

Quantum transmission processes and their entropies

In classical information theory, one of the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov-Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes-Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy are done [10, 3, 8, 39, 15, 44, 9, 27, 28, 2, 19, 45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S- mixing entropy by Ohya [26, 27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S- mixing entropy. In this paper, we discuss the efficiency of information transmission to calculate the mean mutual entropy with respect to the modulated initial states and the connected channel for the quantum dynamical systems.     

Entanglement of a two-level atom papered in a finite trio-coherent state

In this paper, we study the interaction between an effective two-level atom and a three-mode field. The atom and the field are initially in the excited state and finite dimensional trio-coherent state, respectively. For this sytem, we investigate the atomic inversion, the von Neumann entropy, and the atomic Wehrl entropy. We show that there is a connection between all of these quantities. Also, we prove that the atomic Wehrl entropy exhibits a temporal evolution similar to the von Neumann entropy. It is observed that the Stark shift parameter plays an important role on the evolution of these quantities.