Volume 49, Issue 1, January 2008
 ANNOUNCEMENTS


Announcement: Special Issue on “Statistical Mechanics on Random Structures”
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 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


New Bell inequalities for the singlet state: Going beyond the Grothendieck bound
View Description Hide DescriptionContemporary versions of Bell’s argument [Physics (Long Island City, N.Y.)1, 195 (1964)] against local hidden variable (LHV) theories are based on the ClauserHorneShimonyHolt (CHSH) [Phys. Rev. Lett.23, 880 (1969)] inequality and various attempts to generalize it. The amount of violation of these inequalities cannot exceed the bound set by the Grothendieck constants. However, if we go back to the original derivation by Bell and use the perfect anticorrelation embodied in the singlet spin state, we can go beyond these bounds. In this paper, we derive twoparticle Bell inequalities for traceless twooutcome observables, whose violation in the singlet spin state go beyond the Grothendieck constants both for the two and three dimensional cases. Moreover, creating a higher dimensional analog of perfect correlations and applying a recent result of Alon et al. [Invent. Math.163, 499 (2006)], we prove that there are twoparticle Bell inequalities for traceless twooutcome observables whose violation increases to infinity as the dimension and number of measurements grow. Technically, these result are possible because perfect correlations (or anticorrelations) allow us to transport the indices of the inequality from the edges of a bipartite graph to those of the complete graph. Finally, it is shown how to apply these results to mixed Werner states, provided that the noise does not exceed 20%.

Spectra of phase point operators in odd prime dimensions and the extended Clifford group
View Description Hide DescriptionWe analyze the role of the extended Clifford group in classifying the spectra of phase point operators within the framework laid out by [Gibbons et al., Phys. Rev. A70, 062101 (2004)] for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of .

Existence of universal entangler
View Description Hide DescriptionA gate is called an entangler if it transforms some (pure) product states to entangled states. A universal entangler is a gate which transforms all product states to entangled states. In practice, a universal entangler is a very powerful device for generating entanglements, and thus provides important physical resources for accomplishing many tasks in quantum computing and quantum information. This paper demonstrates that a universal entangler always exists except for a degenerated case. Nevertheless, the problem how to find a universal entangler remains open.

The Hahn polynomials in the nonrelativistic and relativistic Coulomb problems
View Description Hide DescriptionWe derive closed formulas for mean values of all powers of in nonrelativistic and relativistic Coulomb problems in terms of the Hahn and Chebyshev polynomials of a discrete variable. A short review on special functions and solution of the Coulomb problems in quantum mechanics is given.

On the degree conjecture for separability of multipartite quantum states
View Description Hide DescriptionWe settle the socalled degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. [Phys. Rev. A73, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag [J. Phys. A40, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomialtime algorithm for completely factorizing any pure multipartite quantum state. By polynomialtime algorithm, we mean that the execution time of this algorithm increases as a polynomial in , where is the number of parts of the quantum system. We give a counterexample to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.

Uncertainty principle with quantum Fisher information
View Description Hide DescriptionIn this paper we prove a lower bound for the determinant of the covariance matrix of quantum mechanical observables, which was conjectured by Gibilisco et al. and has the interpretation of uncertainty. The lower bound is given in terms of the commutator of the state and the observables and quantum Fisher information(generated by an operator monotone function).

Mapping the geometry of the group
View Description Hide DescriptionIn this paper, we present a construction for the compact form of the exceptional Lie group by exponentiating the corresponding Lie algebra, which we realize as the sum of , the derivations of the exceptional Jordan algebra of dimension 3 with octonionic entries, and the right multiplication by the elements of with vanishing trace. Our parametrization is a generalization of the Euler angles for SU(2) and it is based on the fibration of via an subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for . An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the group manifold.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Dualization of the principal sigma model
View Description Hide DescriptionThe firstorder formulation of the principal sigma model with a Lie group target space is performed. By using the dualization of the algebra and the field content of the theory the field equations which are solely written in terms of the field strengths are realized through an extended symmetry algebra parametrization. The structure of this symmetry algebra is derived so that it generates the realization of the field equations in a Bianchi identity of the current derived from the extended parametrization.

Stability of atoms and molecules in an ultrarelativistic ThomasFermiWeizsäcker model
View Description Hide DescriptionWe consider the zero mass limit of a relativistic ThomasFermiWeizsäcker model of atoms and molecules. We find bounds for the critical nuclear charges that ensure stability.
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 GENERAL RELATIVITY AND GRAVITATION


A Riccati equation in radiative stellar collapse
View Description Hide DescriptionWe model the behavior of a relativistic spherically symmetric shearing fluid undergoing gravitational collapse with heat flux. It is demonstrated that the governing equation for the gravitational behavior is a Riccati equation. We show that the Riccati equation admits two classes of new solutions in closed form. We regain particular models, obtained in previous investigations, as special cases. A significant feature of our solutions is the general spatial dependence in the metric functions which allows for a wider study of the physical features of the model, such as the behavior of the causal temperature in inhomogeneous spacetimes.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


HarishChandra modules over the twisted HeisenbergVirasoro algebra
View Description Hide DescriptionIn this paper, we classify all indecomposable HarishChandra modules of the intermediate series over the twisted HeisenbergVirasoro algebra. Meanwhile, some bosonic modules are also studied.
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 STATISTICAL PHYSICS


Generalization of the classical Kramers rate for nonMarkovian open systems out of equilibrium
View Description Hide DescriptionWe analyze the behavior of a Brownian particle moving in a doublewell potential. The escape probability of this particle over the potential barrier from a metastable state toward another state is known as the Kramers problem. In this work, we generalize Kramers’ rate theory to the case of an environment always out of thermodynamic equilibrium reckoning with nonMarkovian effects.
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 METHODS OF MATHEMATICAL PHYSICS


Two types of generalized integrable decompositions and new solitarywave solutions for the modified KadomtsevPetviashvili equation with symbolic computation
View Description Hide DescriptionThe modified KadomtsevPetviashvili (mKP) equation is shown in this paper to be decomposable into the first two solitonequations of the coupled ChenLeeLiu and KaupNewell hierarchies by, respectively, nonlinearizing two sets of symmetry Lax pairs. In these two cases, the decomposed dimensional nonlinear systems both have a couple of different Lax representations, which means that there are two linear systems associated with the mKP equation under the same constraint between the potential and eigenfunctions. For each Lax representation of the decomposed dimensional nonlinear systems, the corresponding Darboux transformation is further constructed such that a series of explicit solutions of the mKP equation can be recursively generated with the assistance of symbolic computation. In illustration, four new families of solitarywave solutions are presented and the relevant stability is analyzed.

On the ternary complex analysis and its applications
View Description Hide DescriptionIn previous works, a possible extension of the complex numbers together with its connected trigonometry was introduced. In the present paper, we focus on the simplest case of ternary complex numbers. Then, some types of holomorphy adapted to the ternary complex numbers and the corresponding results upon integration of differential forms are given. Several physical applications are discussed and, in particular, one type of holomorphic function gives rise to a new form of stationary magnetic field. The movement of a monopoletype object in this field is then studied and shown to be integrable. The monopole scattering in the ternary field is finally studied.

Monomial integrals on the classical groups
View Description Hide DescriptionThis paper presents a powerful method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group by one of the authors, Gorin [J. Math. Phys., 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration methods. The integration formulas are easily implemented in a computer algebra environment, which allows us to compute a given monomial integral very efficiently. The result is always a rational function of the matrix dimension.

Relations between convergence rates in Schatten norms
View Description Hide DescriptionIn quantum estimation theory and quantum tomography, the quantum state obtained by sampling converges to the “true” unknown density matrix under topologies that are different from the natural notion of distance in the space of quantum states, i.e., the trace class norm. In this paper, we address such problem, finding relations between the rates of convergence in the Schatten norms and in the trace class norm.

Infinitedimensional representations of the rotation group and Dirac monopole problem
View Description Hide DescriptionWithin the context of infinitedimensional representations of the rotation group, the Dirac monopole problem is studied in detail. Irreducible infinitedimensional representations, which have been realized in the indefinite metric Hilbert space, are given by linear unbounded operators in infinitedimensional topological spaces, supplied with a weak topology and associated weak convergence. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the socalled topological spin, which is related to the weight of the Dirac string.

Source generation of the DaveyStewartson equation
View Description Hide DescriptionThe “source generation” procedure (SGP) proposed by Hu and Wang [Inverse Probl.22, 1903 (2006)] provides a new way to systematically generate socalled solitonequations with selfconsistent sources. In this paper, we apply this SGP to a DaveyStewartson (DS) equation based on the Hirota bilinear form, producing a system of equations which is called the DS equation with selfconsistent sources (DSESCS). Meanwhile, we obtain the Grammtype determinant solutions to the DSESCS. Since the DS equation is a dimensional integrable generalization of the nonlinear Schrödinger (NLS) equation, the DSESCS may be viewed as a dimensional integrable generalization of the nonlinear Schrödinger equation with selfconsistent sources. These results indicate the commutativity of source generation procedure and dimensional integrable generalizations for the NLS equation.

Applications of the generalized Lüders theorem
View Description Hide DescriptionIn this paper, we prove that the main results in a more recent paper [S. Gudder, “Duality quantum computers and quantum operations,” Int. J. Theor. Phys. (in press)] also hold on a separable complex Hilbert space. The proofs in the present paper have been completed by using the block operator technique which are different from that in Gudder’s paper on the unitary freedom theorem of the finite dimensional setting.
