Theory of a Pair of Quantum Bits

In this paper we present a thorough study of the theory of a pair of qubits, whose Hilbert space can be identified with ^{2 2}. Given an hermitian operator of trace 1 in ^{2 2} we focus on the following Problems: Problem 1: Find conditions that guarantee that is a state, that is, positive semidefinite. Problem 2: Find conditions that guarantee that a given state is separable, or that is a convex combination of products of one-particle states. The language we develop for our investigation makes use of the observation that ^{2 2} carries representations of the special unitary group SU(2) in two dimensions and of the direct product of this group by itself. We introduce a new type of observable called Bell observable (section 5) and a new measure of entanglement called concurrence, which is closely related to the concurrence introduced by Wootters (Physical Review Letters (1998) 80, 2245–2248) (section 8). The work has been inspired by the works of Wootters (Physical Review Letters (1997) 78, 5022–5025; Physical Review Letters (1998) 80, 2245–2248) and members of the Horodecki family (cf Horodecki and Horodecki, Physical Review A (1996) 54, 1838–1843; Horodecki et al., Physics Letters A (1996a) 223, 1–8; Physics Letters A (1996b) 222, 21–25) and reproduces some of their results.     

Hölder Regularity of Integrated Density of States for the Almost Mathieu Operator in a Perturbative Regime

Consider the Almost Mathieu operator H = cos 2(k +)+ on the lattice. It is shown that for large , the integrated density of states is Hölder continuous of exponent < . This result gives a precise version in the perturbative regime of recent work by M. Goldstein and W. Schlag on Hölder regularity of the integrated density of states for 1D quasi-periodic lattice Schrödinger operators, assuming positivity of the Lyapunov exponent (and proven by different means). Our approach provides also a new way to control Green's functions, in the spirit of the author's work in KAM theory. It is by no means restricted to the cosine-potential and extends to band operators.     

Planck—Wheeler Quantum Foam as White Noise: Metric Diffusion and Congruence Focussing for Fluctuating Spacetime Geometry

It is expected that quantum effects endow spacetime with stochastic properties near the Planck scale as exemplified by random fluctuations of the metric, usually referred to as spacetime foam or geometrodynamics. In this paper, a methodology is presented for incorporating Planck scale stochastic effects and corrections into general relativity within the ADM formalism, by coupling the Riemann 3-metric to white noise. The ADM—Cauchy evolution of a Riemann 3-metric h _ij (t) induced on spacelike hypersurface C(t) can be interpreted within pure general relativity as a smooth geodesic flow in superspace, whose points consist of equivalence classes of 3-metrics. Coupling white noise to h _ij gives Langevin stochastic differential equations for the Cauchy evolution of h _ij, which is now a Brownian motion or diffusion in superspace. A fluctuation h_ij away from h _ij is considered to be related to h _ij by elements of the diffeomorphism group diff(C). Hydrodynamical Fokker—Planck continuity equations are formulated describing the stochastic Cauchy evolution of h _ij as a probability flow. The Cauchy invariant or equilibrium solution gives a stationary probability distribution of fluctuations peaked around the deterministic metric. By selecting a physically viable ansatz for the scale dependent diffusion coefficient, one reproduces the Wheeler uncertainty relation for the metric fluctuations of quantum geometrodynamics. Treating h _ij as a random variable, a non-linear Raychaudhuri—Langevin equation is derived describing the geometro-hydrodynamics of a congruence of fluid or dust matter propagating on the stochastic spacetime. For an initially converging congruence >0 at s the singularity =– at future proper time s=3/||$, which is expected in general relativity, is now smeared out near the Planck scale. Proper time s can be extended indefinitely (s) so that intrinsic metric fluctuations can restore geodesic completeness although the geodesics remain trapped for all time: although a singularity can be removed the collapsing matter still creates a black hole. A Fokker—Planck formulation also gives zero probability that – for s. Essentially, the short distance stochastic corrections to the deterministic equations of general relativity can remove pathologies such as singularities, conjugate points and geodesic incompleteness.     

Change in primary extinction during decomposition of supersaturated solid solution II. Al-Cu 4% alloy-formation of G.P. zones

The change in integrated intensity of the (200) reflections of a solid solution during the formation of G.P. zones was measured and compared with the change in the character of the diffuse streaks corresponding to them. It was found that the. formation of G.P. zones does not lead to a decrease in primary extinction despite the great changes in the distribution of the copper atoms. It was shown that the formation of a precipitate accompanied by the formation of crystallographically incoherent boundaries greatly decreases the primary extinction.

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. II. 1-u 4%: [. . ]

(200) . . , . , . . , . , , , .

     

Semi-classical asymptotics in solid state physics

This article studies the Schrödinger equation for an electron in a lattice of ions with an external magnetic field. In a suitable physical scaling the ionic potential becomes rapidly oscillating, and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from homogenization. For the time-dependent Schrödinger equation this construction yields wave packets which follow the trajectories of the semiclassical model. For the time-independent equation one gets asymptotic eigenfunctions (or quasimodes) for the energy levels predicted by Onsager's relation.     

Singularities in Gravitational Collapse with Radial Pressure

We analyze spherical dust collapse with non-vanishing radial pressure, II, and vanishing tangential stresses. Considering a barotropic equation of state, II = , we obtain an analytical solution in closed form—which is exact for = –1, 0, and approximate otherwise—near the center of symmetry (where the curvature singularity forms). We study the formation, visibility, and curvature strength of singularities in the resulting spacetime. We find that visible, Tipler strong singularities can develop from generic initial data. Radial pressure alters the spectrum of possible endstates for collapse, increasing the parameter space region that contains no visible singularities, but cannot by itself prevent the formation of visible singularities for sufficiently low values of the energy density. Known results from pressureless dust are recovered in the = 0 limit.     

Multifractal Power Law Distributions: Negative and Critical Dimensions and Other “Anomalies,” Explained by a Simple Example

Divergence of high moments and dimension of the carrier is the subtitle of Mandelbrot's 1974 seed paper on random multifractals. The key words divergence and dimension met very different fates. Dimension expanded into a multifractal formalism based on an exponent and a function f(). An excellent exposition in Halsey et al. 1986 helped this formalism flourish. But it does not allow divergent high moments and the related inequalities f()<0 and <0. As a result, those possibilities did not flourish. Now their time has come for diverse reasons. The broad 1974 definitions of and f allow <0 and f()<0, but the original presentation demanded to be both developed and simplified. This paper shows that both multifractal anomalies occur in a very simple example, which has been crafted for this purpose. This example predicts the power law distribution. It generalizes and f() beyond their usual roles of being a Hölder exponent and a Hausdorff dimension. The effect is to allow either f or both f and to be negative, and the apparent anomalies are made into sources of new important information. In addition, this paper substantially clarifies the subtle way in which randomness manifests itself in multifractals.     

Besov spaces and the multifractal hypothesis

Parisi and Frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis, i.e., they conjecture that the velocity field has local Hölder exponents in a range [h _min,h _max], with exponents <h occurring on a setS(h) with a fractal dimensionD(h). Heuristic reasoning led them to an expression for the scaling exponentz _p ofpth order as the Legendre transform of the codimensiond-D(h). We show here that a part of the multifractal hypothesis is correct under even weaker assumptions: namely, if the velocity field hasL ^p -mean Hölder indexs, i.e., if it lies in the Besov spaceB _p ^s, , then local Hölder regularity is satisfied. Ifs<d/p, then the hypothesis is true in a generalized sense of Hölder space with negative exponents and we discuss the proper definition of local Hölder classes of negative index. Finally, if a certain box-counting dimension exists, then the Legendre transform of its codimension gives the scaling exponentz _p , and, more generally, the maximal Besov index of order,p, ass _p =z _p /p. Our method of proof is derived from a recent paper of S. Jaffard using compactly-supported, orthonormal wavelet bases and gives an extension of his results. We discuss implications of the theorems for ensemble-average scaling and fluid turbulence.     

On the theory of ferromagnetic thin films

As is known, the second approximation in the calculation of the partition function by the traces method of ferromagnetic thin films gives wrong results for the coordination number equal to eight. In order to obtain correct results even for this case, the third order approximation of the partition function is developed and thus the magnetic properties of body-centred cubic iron thin films are studied. The dependence of the Curie temperature on the thickness, for different values of the ratio between the anisotropy constant and the exchange energy between two neighbours, is discussed. A value can be chosen for this ratio such that the thin film becomes ferromagnetic only for a thickness greater than a definite value.

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, . , , (Fe). . , - .

     

Induced geometry and confinement

A classical, Poincaré invariant dynamical system is developed which contains, besides the natural metric _v , an induced metricg _v that is generated by a real scalar dynamical field. It is shown that scalar fields whose dynamics are governed by the induced metric can be consistently introduced. Also, point particles which follow timelike quasi-geodesic trajectories can be introduced. The reaction forces acting ong _v due to the presence of these fields and particles are computed. A discussion of causality and geometrical confinement is given.     

A dimension formula for Bernoulli convolutions

We present a dynamical approach to the study of the singularity of infinitely convolved Bernoulli measuresv , for the golden section. We introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square, the fat baker's transformation. Our approach strongly relies on the Markov structure of the underlying dynamical system. Indeed, if =golden mean, the fat baker's transformation has a very simple Markov coding. The ambiguity (of order two) of this coding, which appears when projecting on the line, due to passages for the central, overlapping zone, can be expressed by means of products of matrices (of order two). This product has a Markov distribution inherited by the Markov structure of the map. The dimension of the projected measure is therefore associated to the growth of this product; our dimension formula appears in a natural way as a version of the Furstenberg-Guivarch formula. Our technique provides an explicit dimension formula and, most important, provides a formalism well suited for the multifractal analysis of this measure, as we will show in a forthcoming paper.     

Effect of electrolytically deposited nickel films on the magnetic properties of iron-silicon alloy

The effect of electrolytically deposited nickel films on the magnetostriction, coercive force, residual induction, and the position with respect to each other of the magnetic susceptibility maxima _max. _max of specimens of iron-silicon alloy is investigated. It is shown that elastic extensions lead to a reduction in the spacing between the maxima _max, _max , and for certain loads this spacing disappears and only one maximum is observed.     

Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian

It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H _mn= _{m, n+1}+ _{m, n–1}+ _{m, n} [(n+1)]–[n]) where =(5–1)/2 and [·] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzero, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.On leave from the Central Research Institute for Physics, Budapest, Hungary.     

Two singular integrals containing a partial wave of the two-body coulomb T-matrix

The operatorsT _C,l E+i0)[–G ₀(E+i0)]^1–i andT _C,l(E+i)G ₀[–iG ₀(E+i)]ⁱ acting on spaces of Hölder continuous, differentiable and analytic functions are investigated. The results of their action are expressed in terms of explicit singular factors and terms and Hölder (differentiable, analytic) functions. The most singular part of these operators is shown to be determined by a simple functional.     

On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state

It is pointed out that chemical reactions which show an absorbing stationary state in the master-equation approach (e.g. Schlögl's first reaction) exhibit nevertheless a second order phase transition in non-zero dimensional macroscopic systems. The relation to Reggeon field theory is given more directly than by Grassberger et al. using the functional integral formalism of statistical dynamics. As a new result the correlation length exponent and the order parameter exponent are found toO(²) in an -expansion around the upper critical dimensiond _c=4.     

Critical circle maps near bifurcation

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 1/3. Moreover, the rotation number is a Hölder continuous function of the parameter.Partially supported by KBN grant Iteracje i Fraktale #210909101.Partially supported by NSF Grant #DMS-9206793 and the Sloan Research Fellowship.     

Dynamics of anisotropic cosmological model with ultrarelativistic matter,magnetic field,and fluxes of free isotropic particles

Within the framework of general relativity a dynamics of homogeneous anistropic axially symmetric model of the Bianchi type I is considered for the case when sources of gravitational field are ultrarelativistic matter, homogeneous magnetic field, and fluxes of free particles. Qualitative analysis of the field equations on a phase plane is given. All solutions of a considered type for large values of proper time asymptotically approach the flat Friedmann model while the value of energy density of free particles approaches the double value of magnetic field energy density. Near a singular state the solution exhibits oscillating behavior with successive interchange of Kasner singularities of pancake-like and filament-like types. It is also shown that in the absence of matter a solution retains its character.     

Towards a generalized distribution formalism for gauge quantum fields

We prove that the distributions defined on the Gelfand-Shilov spacesS with < 1 and, hence, more singular than hyperfunctions, retain the angular localizability property. Specifically, they have uniquely determined support cones. This result enables one to develop a distribution-theoretic technique suitable for the consistent treatment of quantum fields with arbitrarily singular ultraviolet and infrared behavior. The proof covering the most general and difficult case = 0 is based on the use of the theory of plurisubharmonic functions and Hörmander'sL ²-estimates.This work was supported in part by a Soros Humanitarian Foundation Grant awarded by the American Physical Society.     

A Convergence exponent for multidimensional continued-fraction algorithms

We study a convergence exponent of multidimensional continued-fraction algorithms (MCFAs). We provide a dynamical systems interpretation for this exponent, then express a general relation for the exponent in terms of the Kolmogorov-Sinai (KS) entropy and smallest eigenvalue of the associated shift map. We consider the case of approximating two irrationals and demonstrate the numerical method for using the smallest eigenvalue and entropy to evaluate for several MCFAs, including Jacobi-Perron and GMA (generalized mediant algorithm). On very general grounds, the bounds for this exponent (for two irrationals) are 13/2=1.5. The upper bound is attained for algorithms with best approximation properties. We find _GMA=1.387 and _JP=1.374, as well as the values for the KS entropy and Oseledec eigenvalues.     

Remarks on the modular operator and local observables

In this paper we give a characterization of the modular group of a von Neumann algebra , with a cyclic and separating vector, which provides at the same time a necessary and sufficient condition so that two von Neumann algebras ₁ and ₂, such that ₁₂, are the mutual commutants, i.e. ₁=₂.An application is made to the duality property in Quantum Field Theory, and we give a sufficient condition for PCT invariance in a theory of local observables.Partially supported by C.N.R.