Volume 49, Issue 10, October 2008
 ANNOUNCEMENTS


Announcement: Special issue on “Integrable Quantum Systems and Solvable Statistical Mechanics Models”
View Description Hide Description  Top

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Asymptotic performance of optimal state estimation in qubit system
View Description Hide DescriptionWe derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.

The effect of classical noise on a quantum twolevel system
View Description Hide DescriptionWe consider a quantum twolevel system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the offdiagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein–Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate onedimensional diffusionequation for the transition probabilities. The proofs use both spectraltheoretic and probabilistic methods.

On the divine clockwork: The spectral gap for the correspondence limit of the Nelson diffusion generator for the atomic elliptic state
View Description Hide DescriptionThe correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane . We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincaré inequalities, both of which are used in our proof of the existence of the spectral gap.

The structure of degradable quantum channels
View Description Hide DescriptionDegradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum informationtheory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or antidegradable, and present a complete description of antidegradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions ( and , respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of . Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a nonorthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension . Indeed, one can have . These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.

Lower spectral branches of a spinboson model
View Description Hide DescriptionWe study the structure of the spectrum of a twolevel quantum system weakly coupled to a boson field (spinboson model). Our analysis allows to avoid the cutoff in the number of bosons, if their spectrum is bounded below by a positive constant. We show that, for small coupling constant, the lower part of the spectrum of the spinboson Hamiltonian contains (one or two) isolated eigenvalues and (one or two) manifolds of atom boson states indexed by the boson momentum . The dispersion laws and generalized eigenfunctions of the latter are calculated.

Adiabatically switchedon electrical bias and the Landauer–Büttiker formula
View Description Hide DescriptionConsider a three dimensional system which looks like a cross connected pipe system, i.e., a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We finally prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer–Büttiker type formula.

Additivity and distinguishability of random unitary channels
View Description Hide DescriptionA random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is as hard as the problem of distinguishing general mixedstate circuits, which is complete for the class of problems having quantum interactive proof systems.

Nine theorems on the unification of quantum mechanics and relativity
View Description Hide DescriptionA mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space of functions of four variables furnished with an additional indefinite inner product invariant under Poincaré transformations is introduced. For a class of functions in that are well localized in the time variable, the usual formalism of nonrelativistic quantum mechanics is derived. In particular, the interference in time for these functions is suppressed; a motion in becomes the usual Schrödinger evolution with as a parameter. The relativistic invariance of the construction is proved. The usual theory of relativity on Minkowski spacetime is shown to be “isometrically and equivariantly embedded” into . That is, classical spacetime is isometrically embedded into , Poincaré transformations have unique extensions to isomorphisms of , and the embedding commutes with Poincaré transformations.

Discrete approximation of quantum stochastic models
View Description Hide DescriptionWe develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noisemodel and does not require boundedness of the limit coefficients.

Classical limit for semirelativistic Hartree systems
View Description Hide DescriptionWe consider the threedimensional semirelativistic Hartree model for fast quantum mechanical particles moving in a selfconsistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove by using Wigner transformation techniques that its classical limit yields the well known relativistic Vlasov–Poisson system. The result holds for the case of attractive and repulsive meanfield interactions, with an additional size constraint in the attractive case.

The O(1)Kepler problems
View Description Hide DescriptionLet be an integer. To each irreducible representation of O(1), an O(1)Kepler problem in dimension is constructed and analyzed. This system is super integrable and when it is equivalent to a generalized MICZ (McIntoshCisnerosZwanziger)Kepler problem in dimension 2. The dynamical symmetry group of this system is with the Hilbert space of bound states being the unitary highest weight representation of with highest weight which occurs at the rightmost nontrivial reduction point in the Enright–Howe–Wallach classification diagram for the unitary highest weight modules. (Here or 1 depending on whether is trivial or not.) Furthermore, it is shown that the correspondence is the theta correspondence for dual pair .

Weyl quantization of fractional derivatives
View Description Hide DescriptionThe quantum analogs of the derivatives with respect to coordinates and momenta are commutators with operators and . We consider quantum analogs of fractional Riemann–Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann–Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for analytic functions. To define a quantum analog of the fractional Liouville derivative, which is defined on the real axis, we can use the representation of the Weyl quantization by the Fourier transformation.
 Top

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Stochastic quantization for complex actions
View Description Hide DescriptionWe use the stochastic quantization method to study systems with complex valued path integral weights. We assume a Langevin equation with a memory kernel and Einstein’s relations with colored noise. The equilibrium solution of this nonMarkovian Langevin equation is analyzed. We show that for a large class of elliptic nonHermitian operators acting on scalar functions on Euclidean space, which define different models in quantum field theory, converge to an equilibrium state in the asymptotic limit of the Markov parameter . Moreover, as we expected, we obtain the Schwinger functions of the theory.

Super Schrödinger algebra in AdS/CFT
View Description Hide DescriptionWe discuss (extended) superSchrödinger algebras obtained as subalgebras of the superconformal algebra. The Schrödinger algebra with two spatial dimensions can be embedded into so(4,2). In the superconformal case the embedded algebra may be enhanced to the socalled superSchrödinger algebra. In fact, we find an extended superSchrödinger subalgebra of . It contains 24 supercharges (i.e., 3/4 of the original supersymmetries) and the generators of so(6), as well as the generators of the original Schrödinger algebra. In particular, the 24 supercharges come from 16 rigid supersymmetries and half of 16 superconformal ones. Moreover, this superalgebra contains a smaller superSchrödinger subalgebra, which is a supersymmetric extension of the original Schrödinger algebra and so(6) by eight supercharges (half of 16 rigid supersymmetries). It is still a subalgebra even if there are no so(6) generators. We also discuss superSchrödinger subalgebras of the superconformal algebras, and , and find super Schrödinger subalgebras in the same way.
 Top

 GENERAL RELATIVITY AND GRAVITATION


On the embedding of spacetime in fivedimensional Weyl spaces
View Description Hide DescriptionWe revisit Weyl geometry in the context of recent higherdimensional theories of spacetime. After introducing the Weyl theory in a modern geometrical language we present some results that represent extensions of Riemannian theorems. We consider the theory of local embeddings and submanifolds in the context of Weyl geometries and show how a Riemannian spacetime may be locally and isometrically embedded in a Weyl bulk. We discuss the problem of classical confinement and the stability of motion of particles and photons in the neighborhood of branes for the case when the Weyl bulk has the geometry of a warped product space. We show how the confinement and stability properties of geodesics near the brane may be affected by the Weyl field. We construct a classical analog of quantum confinement inspired in theoreticalfield models by considering a Weyl scalar field which depends only on the extra coordinate.
 Top

 DYNAMICAL SYSTEMS


Analytic integrability of a Chua system
View Description Hide DescriptionWe consider the system, , , where and are parameters and , , , and . We analyze the existence of local and global analytic first integrals.

On the classification of Darboux integrable chains
View Description Hide DescriptionWe study a differentialdifference equation of the form with unknown depending on and . The equation is called a Darboux integrable if there exist functions (called an integral) and (called an integral), both of a finite number of variables , such that and , where is the operator of total differentiation with respect to and is the shift operator: . The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial integrals is given in the case when the function is of the special form .

Approximation of center manifolds on the renormalization group method
View Description Hide DescriptionThe renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining approximate solutions. This article shows that the RG method is effectual for obtaining an approximate center manifold and an approximate flow on it when applied to equations having a center manifold.

Hydrodynamic type integrable equations on a segment and a halfline
View Description Hide DescriptionThe concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions in multifield systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semiline are presented.
