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Dec 2005

Volume 46, Issue 12, Articles (12xxxx)

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Constraints on the mixing of bipartite states

Hao Chen

J. Math. Phys. 46, 122101 (2005); http://dx.doi.org/10.1063/1.2138048 (7 pages)

Online Publication Date: 13 December 2005

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A mixed state may be represented in many different ways as a mixture of pure states ρ = ∑pi∣ψi〉〈ψi∣. The mixing problem in quantum mechanics asks the characterization of the probability distribution (pi) and the mixed states (ρi) such that ρ = ∑piρi for any given mixed state ρ. Some constraints based on eigenvalues of the mixed states are established in uni-party case [see Nielsen, Phys. Rev. A. 63, 052308 (2000) , 63, 022144 (2000) , Nielsern and Vidal Quantum Inf. Comput. 1 76 (2001) ]. We develop some new invariant sets for bipartite mixed states under local unitary operations, which are independent of eigenvalues, and prove some strong constraints based on these invariant sets for the mixing problem in bipartite case. This exhibits a remarkable difference from the uni-party case.
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03.65.Fd Algebraic methods
02.50.Cw Probability theory
02.50.Ng Distribution theory and Monte Carlo studies
02.10.Ud Linear algebra

Ladder operators and coherent states for the Jaynes-Cummings model in the rotating-wave approximation

V. Hussin and L. M. Nieto

J. Math. Phys. 46, 122102 (2005); http://dx.doi.org/10.1063/1.2137718 (21 pages) | Cited 16 times

Online Publication Date: 14 December 2005

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Using algebraic techniques, we realize a systematic search of different types of ladder operators for the Jaynes-Cummings model in the rotating-wave approximation. The link between our results and previous studies on the diagonalization of the associated Hamiltonian is established. Using some of the ladder operators obtained before, examples are given on the possibility of constructing a variety of interesting coherent states for this Hamiltonian.
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42.50.Ar Photon statistics and coherence theory
03.65.Fd Algebraic methods

Quantitative estimates on the enhanced binding for the Pauli-Fierz operator

Jean-Marie Barbaroux, Helmut Linde, and Semjon Vugalter

J. Math. Phys. 46, 122103 (2005); http://dx.doi.org/10.1063/1.2142835 (11 pages) | Cited 1 time

Online Publication Date: 16 December 2005

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For a quantum particle interacting with a short-range potential, we estimate from below the shift of its binding threshold, which is due to the particle interaction with a quantized radiation field.
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03.65.Ge Solutions of wave equations: bound states
12.20.Ds Specific calculations
02.30.Tb Operator theory

Analytic Coulomb matrix elements in a three-dimensional geometry

Jaime Zaratiegui García

J. Math. Phys. 46, 122104 (2005); http://dx.doi.org/10.1063/1.2146187 (5 pages)

Online Publication Date: 23 December 2005

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Using a complete basis set we have obtained an analytic expression for the matrix elements of the Coulomb interaction. These matrix elements are written in a closed form. We have used the basis set of the three-dimensional isotropic quantum harmonic oscillator in order to develop our calculations, which can be useful when treating interactions in localized systems.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods
02.10.Yn Matrix theory
02.40.-k Geometry, differential geometry, and topology

A combinatorial approach for studying local operations and classical communication transformations of multipartite states

Sudhir Kumar Singh, Sudebkumar Prasant Pal, Somesh Kumar, and R. Srikanth

J. Math. Phys. 46, 122105 (2005); http://dx.doi.org/10.1063/1.2142840 (22 pages) | Cited 1 time

Online Publication Date: 27 December 2005

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We develop graph theoretic methods for analyzing maximally entangled pure states distributed between a number of different parties. We introduce a technique called bicolored merging, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication. We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among n agents. Further, we discuss a new graph theoretic metric on a class of multipartite states, and its implications.
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03.67.Mn Entanglement measures, witnesses, and other characterizations
03.67.Hk Quantum communication
02.10.Ox Combinatorics; graph theory

Generalized coherent states and the control of quantum systems

Holger Teismann

J. Math. Phys. 46, 122106 (2005); http://dx.doi.org/10.1063/1.2138051 (13 pages)

Online Publication Date: 30 December 2005

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The control problem for linear and nonlinear Schrödinger equations is considered. The controls are given by applying a spatially homogeneous field or varying the frequency of a quadratic trapping potential. It is demonstrated that the existence of (exact or approximate) coherent-state-type solutions may severely limit the degree to which the system can be controlled.
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03.65.Ge Solutions of wave equations: bound states
02.30.-f Function theory, analysis

Decomposition of time-covariant operations on quantum systems with continuous and∕or discrete energy spectrum

Dominik Janzing

J. Math. Phys. 46, 122107 (2005); http://dx.doi.org/10.1063/1.2142839 (20 pages) | Cited 1 time

Online Publication Date: 30 December 2005

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Every completely positive map G that commutes with the Hamiltonian time evolution is an integral or sum over (densely defined) CP-maps Gσ where σ is the energy that is transferred to or taken from the environment. If the spectrum is non-degenerate each Gσ is a dephasing channel followed by an energy shift. The dephasing is given by the Hadamard product of the density operator with a (formally defined) positive operator. The Kraus operator of the energy shift is a partial isometry which defines a translation on math with respect to a non-translation-invariant measure. As an example, this decomposition is explicitly calculated for the rotation invariant Gaussian channel on a single mode. The question of under what conditions a covariant channel destroys superpositions between mutually orthogonal states on the same orbit is addressed. For channels which allow mutually orthogonal output states on the same orbit, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (Gσ).
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03.65.Ge Solutions of wave equations: bound states
02.30.Uu Integral transforms
02.30.Rz Integral equations

A de Finetti representation for finite symmetric quantum states

Robert König and Renato Renner

J. Math. Phys. 46, 122108 (2005); http://dx.doi.org/10.1063/1.2146188 (23 pages) | Cited 8 times

Online Publication Date: 30 December 2005

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Consider a symmetric quantum state on an n-fold product space, that is, the state is invariant under permutations of the n subsystems. We show that, conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems, the state in the remaining subsystems is close to having product form. This immediately generalizes the so-called de Finetti representation to the case of finite symmetric quantum states.
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03.65.Ta Foundations of quantum mechanics; measurement theory
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Non-Abelian Chern-Simons action is topological invariant on 3 simple knot

Tieyan Si

J. Math. Phys. 46, 122301 (2005); http://dx.doi.org/10.1063/1.2137721 (4 pages)

Online Publication Date: 12 December 2005

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Under SU(2) gauge transformation, the non-Abelian Chern-Simons action is invariant on a class of three dimensional manifold—3 simple knot.
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11.15.-q Gauge field theories
02.40.Pc General topology
11.30.Ly Other internal and higher symmetries

On Chern-Simons theory with an inhomogeneous gauge group and BF theory knot invariants

Gad Naot

J. Math. Phys. 46, 122302 (2005); http://dx.doi.org/10.1063/1.2146190 (13 pages)

Online Publication Date: 28 December 2005

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We study the Chern-Simons topological quantum field theory with an inhomogeneous gauge group, a non-semi-simple group obtained from a semisimple one by taking its semidirect product with its Lie algebra. We find that the standard knot observable (i.e., trace of the holonomy along the knot) essentially vanishes, and yet, the non-semi-simplicity of the gauge group allows us to consider a class of unorthodox observables which breaks gauge invariance at one point and leads to a nontrivial theory on long knots in math3. We have two main morals. (1) In the non-semi-simple case there is more to observe in Chern-Simons theory. There might be other interesting non-semi-simple gauge groups to study in this context beyond our example. (2) In the case of an inhomogeneous gauge group, we find that Chern-Simons theory with the unorthodox observable is actually the same as three-dimensional BF theory with the Cattaneo-Cotta-Ramusino-Martellini knot observable. This leads to a simplification of their results and enables us to generalize and solve a problem they posed regarding the relation between BF theory and the Alexander-Conway polynomial. We prove that the most general knot invariant coming from pure BF topological quantum field theory is in the algebra generated by the coefficients of the Alexander-Conway polynomial.
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11.15.Kc Classical and semiclassical techniques
02.20.Sv Lie algebras of Lie groups
02.10.Ud Linear algebra
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Noncommutative unification of general relativity and quantum mechanics

Michael Heller, Leszek Pysiak, and Wiesław Sasin

J. Math. Phys. 46, 122501 (2005); http://dx.doi.org/10.1063/1.2137720 (15 pages) | Cited 6 times

Online Publication Date: 9 December 2005

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We present a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry is developed in terms of a noncommutative algebra A which is defined on a transformation groupoid Γ given by the action of a noncompact group G on the total space E of a principal fiber bundle over space-time M. The case is important since to obtain physical effects predicted by the model we should assume that G is a Lorentz group or some of its representations. We show that the generalized Einstein equation of the model has the form of the eigenvalue equation for the generalized Ricci operator, and all relevant operators in the quantum sector of the model are random operators; we study their dynamics. We also show that the model correctly reproduces general relativity and the usual quantum mechanics. It is interesting that the latter is recovered by performing the measurement of any observable. In the act of such a measurement the model “collapses” to the usual quantum mechanics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
03.65.Fd Algebraic methods
02.40.Gh Noncommutative geometry
02.10.Ud Linear algebra

Causal sites as quantum geometry

J. Daniel Christensen and Louis Crane

J. Math. Phys. 46, 122502 (2005); http://dx.doi.org/10.1063/1.2138043 (17 pages) | Cited 2 times

Online Publication Date: 15 December 2005

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We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural “tangent 2-bundle,” analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well.
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04.60.Pp Loop quantum gravity, quantum geometry, spin foams
04.20.Gz Spacetime topology, causal structure, spinor structure
02.40.-k Geometry, differential geometry, and topology

Universal homogeneous causal sets

Manfred Droste

J. Math. Phys. 46, 122503 (2005); http://dx.doi.org/10.1063/1.2147607 (10 pages) | Cited 1 time

Online Publication Date: 30 December 2005

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Causal sets are particular partially ordered sets which have been proposed as a basic model for discrete space-time in quantum gravity. We show that the class math of all countable past-finite causal sets contains a unique causal set (U, ⩽ ) which is universal (i.e., any member of math can be embedded into (U, ⩽ )) and homogeneous (i.e., (U, ⩽ ) has maximal degree of symmetry). Moreover, (U, ⩽ ) can be constructed both probabilistically and explicitly.
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04.60.-m Quantum gravity
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Nonstandard connections in k-cosympletic field theory

Miguel-C. Muñoz-Lecanda, Modesto Salgado, and Silvia Vilariño

J. Math. Phys. 46, 122901 (2005); http://dx.doi.org/10.1063/1.2146191 (25 pages) | Cited 1 time

Online Publication Date: 28 December 2005

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In the jet-bundle description of time-dependent mechanics there are some elements, such as the Lagrangian energy and the construction of the Hamiltonian formalism, which require the prior choice of a connection. This situation is analyzed by Echeverría-Enríquez et al. [J. Phys. A 28, 5553–5567 (1995) ]. The aim of this paper is to extend the results in that paper to first order field theory, using the k-cosymplectic formalism described by de León and co-workers [ J. Math. Phys. 39, 876–893 (1998) ; 42, 2092–2104 (2001) ]. If the trivial configuration bundle of a Lagrangian system is endowed with one connection, different from the trivial one given by the product structure, we study the consequences on the geometric elements of the theory, the dynamical equations and the variational principles.
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03.50.-z Classical field theories
45.20.Jj Lagrangian and Hamiltonian mechanics
45.10.Db Variational and optimization methods
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On the sharpness of the zero-entropy-density conjecture

S. Farkas and Z. Zimborás

J. Math. Phys. 46, 123301 (2005); http://dx.doi.org/10.1063/1.2138047 (8 pages) | Cited 6 times

Online Publication Date: 15 December 2005

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The zero-entropy-density conjecture states that the entropy density defined as s ≔ limN→∞SN/N vanishes for all translation-invariant pure states on the spin chain. Or equivalently, SN, the von Neumann entropy of such a state restricted to N consecutive spins, is sublinear. In this paper it is proved that this conjecture cannot be sharpened, i.e., translation-invariant states give rise to arbitrary fast sublinear entropy growth. The proof is constructive, and is based on a class of states derived from quasifree states on a CAR algebra. The question whether the entropy growth of pure quasifree states can be arbitrary fast sublinear was first raised by Fannes et al. [J. Math. Phys. 44, 6005 (2003)] . In addition to the main theorem it is also shown that the entropy asymptotics of all pure shift-invariant nontrivial quasifree states is at least logarithmic.
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05.30.-d Quantum statistical mechanics
05.70.Ce Thermodynamic functions and equations of state
02.10.-v Logic, set theory, and algebra

Extensive ground state entropy in supersymmetric lattice models

Hendrik van Eerten

J. Math. Phys. 46, 123302 (2005); http://dx.doi.org/10.1063/1.2142836 (8 pages) | Cited 6 times

Online Publication Date: 20 December 2005

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We present the result of calculations of the Witten index for a supersymmetric lattice model on lattices of various type and size. Because the model remains supersymmetric at finite lattice size, the Witten index can be calculated using row-to-row transfer matrices and the calculations are similar to calculations of the partition function at negative activity −1. The Witten index provides a lower bound on the number of ground states. We find strong numerical evidence that the Witten index grows exponentially with the number of sites of the lattice, implying that the model has extensive entropy in the ground state.
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11.30.Pb Supersymmetry
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Nonlinear diffusion equation, Tsallis formalism and exact solutions

P. C. Assis, L. R. da Silva, E. K. Lenzi, L. C. Malacarne, and R. S. Mendes

J. Math. Phys. 46, 123303 (2005); http://dx.doi.org/10.1063/1.2142838 (7 pages) | Cited 7 times

Online Publication Date: 23 December 2005

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We address this work to analyze a nonlinear diffusion equation in the presence of an absorption term taking external forces and spatial time-dependent diffusion coefficient into account. The nonlinear terms present in this equation are due to a nonlinear generalization of the Darcy law and the presence of an absorbent (source) term. We obtain new exact solutions and investigate nonlinear effects produced on the solutions by these terms. We also connect the results found here within the Tsallis formalism.
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05.60.-k Transport processes
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Eigenvalues of Casimir invariants for Uq[osp(m∣n)]

K. A. Dancer, M. D. Gould, and J. Links

J. Math. Phys. 46, 123501 (2005); http://dx.doi.org/10.1063/1.2137712 (20 pages) | Cited 2 times

Online Publication Date: 9 December 2005

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For each quantum superalgebra Uq[osp(m∣n)] with m>2, an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary irreducible highest weight module is also calculated.
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03.65.Fd Algebraic methods
02.10.Ud Linear algebra

Multiple nodal bound states for a quasilinear Schrödinger equation

Jianqing Chen and Boling Guo

J. Math. Phys. 46, 123502 (2005); http://dx.doi.org/10.1063/1.2138045 (11 pages) | Cited 2 times

Online Publication Date: 12 December 2005

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Nehari techniques are used to prove the existence of multiple (indeed infinitely many) nodal type bound states for the quasilinear Schrödinger equation izt−V0z+z″+q(x)∣z∣2z+k(∣z∣2)″z = 0 with prescribed number of nodes.
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03.65.Ge Solutions of wave equations: bound states

Invariant noncommutative connections

Thierry Masson and Emmanuel Serié

J. Math. Phys. 46, 123503 (2005); http://dx.doi.org/10.1063/1.2131206 (25 pages) | Cited 4 times

Online Publication Date: 14 December 2005

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In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitly some geometric constructions usually introduced to classify ordinary invariant connections, and we expand them using algebraic objects coming from the noncommutative setting. The main result is that the classification can be performed using a “reduced” algebra, an associated differential calculus and a module over this algebra.
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02.40.Gh Noncommutative geometry
02.10.Ud Linear algebra

Construction of new solutions to the fully nonlinear generalized Camassa-Holm equations by an indirect F function method

Emmanuel Yomba

J. Math. Phys. 46, 123504 (2005); http://dx.doi.org/10.1063/1.2137723 (12 pages) | Cited 10 times

Online Publication Date: 16 December 2005

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An indirect F function method is introduced to solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p). Taking advantage of elliptic equation, this F function is used to map the solutions of the generalized Camassa-Holm equation to those of the elliptic equation. As a result, we can successfully obtain in a unified way and for special values of the parameters of this equation, many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions (soliton, combined soliton solutions, and triangular solutions) as the modulus m is driven to 1 and 0.
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02.30.-f Function theory, analysis

On the spectral L2 conjecture, 3/2-Lieb-Thirring inequality and distributional potentials

Alexei Rybkin

J. Math. Phys. 46, 123505 (2005); http://dx.doi.org/10.1063/1.2142837 (8 pages) | Cited 2 times

Online Publication Date: 20 December 2005

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Let H = −∂x2+V(x) be a properly defined Schrödinger operator on L2(math) with real potentials of the form V(x) = q(x)+p′(x) (the derivative is understood in the distributional sense) with some p,q ∊ L2(math). We prove that the absolutely continuous spectrum of H fills (0,∞) which was previously proven by Deift-Killip for V ∊ L2(math). We also refine the 3/2-Lieb-Thirring inequality.
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03.65.Ge Solutions of wave equations: bound states
02.30.Tb Operator theory
02.30.Rz Integral equations

Framework for potential systems and nonlocal symmetries: Algorithmic approach

George Bluman and Alexei F. Cheviakov

J. Math. Phys. 46, 123506 (2005); http://dx.doi.org/10.1063/1.2142834 (19 pages) | Cited 15 times

Online Publication Date: 22 December 2005

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An algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations (DEs). Each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree. Useful conservation laws play an essential role in the construction of an extended tree. A useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system. Nonlocal symmetries for a given system of DEs can arise from any system in its extended tree. We construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations, and in both cases obtain new nonlocal symmetries. More importantly, due to the equivalence of solution sets, any coordinate-independent method of analysis (qualitative, numerical, perturbation, etc.) can be applied to any system within the tree, and may yield simpler computations and/or results that cannot be obtained when the method is directly applied to the given system.
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05.60.Gg Quantum transport
02.10.Ox Combinatorics; graph theory
02.30.Hq Ordinary differential equations
03.65.-w Quantum mechanics

The short pulse hierarchy

J. C. Brunelli

J. Math. Phys. 46, 123507 (2005); http://dx.doi.org/10.1063/1.2146189 (9 pages) | Cited 26 times

Online Publication Date: 29 December 2005

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We study a new hierarchy of equations containing the short pulse equation, which describes the evolution of very short pulses in nonlinear media, and the elastic beam equation, which describes nonlinear transverse oscillations of elastic beams under tension. We show that the hierarchy of equations is integrable. We obtain the two compatible Hamiltonian structures. We construct an infinite series of both local and nonlocal conserved charges. A Lax description is presented for both systems. For the elastic beam equations we also obtain a nonstandard Lax representation.
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46.25.Cc Theoretical studies
03.65.Ge Solutions of wave equations: bound states
02.30.Ik Integrable systems

Representations of the q-deformed algebras Uq(so3) and Uq(so5) and q-orthogonal polynomials

Alexander Rozenblyum

J. Math. Phys. 46, 123508 (2005); http://dx.doi.org/10.1063/1.2146192 (14 pages)

Online Publication Date: 29 December 2005

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Orthogonal polynomials related to irreducible representations of the classical type of the q-deformed algebras Uq(so3) and Uq(so5) are investigated. The main method consists in the diagonalization of corresponding infinitesimal operators (generators) of representations. For the algebra Uq(so3) this method leads to q-analogs of Krawtchouk polynomials. The properties of these polynomials are considered, the q-difference equation, the recurrence and explicit formulas. For the algebra Uq(so5), the diagonalization process of generators of representations leads to the connection with some class of orthogonal polynomials in two discrete variables. These variables are the so-called q-numbers [n], where [n] = (qn−q−n)/(q−q−1). The introduced polynomials can be considered as two-dimensional q-analogs of Krawtchouk polynomials. The q-difference equation of the Sturm-Liouville type for these polynomials is constructed. The corresponding eigenvalues are investigated including the explicit formulas for their multiplicities. The structure of polynomial solutions is described.
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02.10.De Algebraic structures and number theory
02.60.Lj Ordinary and partial differential equations; boundary value problems
02.10.Ud Linear algebra
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