Volume 51, Issue 9, September 2010
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Maximal rank of extremal marginal tracial states
View Description Hide DescriptionStates on the coupled quantum system whose restrictions to each subsystem are the normalized traces are called marginal tracial states. We investigate extremal marginal tracial states and compute their maximal rank. Diagonal marginal tracial states are also considered.

Continuoustime random walk as a guide to fractional Schrödinger equation
View Description Hide DescriptionWe argue that the continuoustime random walk approach may be a useful guide to extend the Schrödinger equation in order to incorporate nonlocal effects, avoiding the inconsistencies raised by Jeng et al. [J. Math. Phys.51, 062102 (2010)]. As an application, we work out a free particle in a half space, obtaining the time dependent solution by considering an arbitrary initial condition.

An exactly solvable Schrödinger equation with finite positive positiondependent effective mass
View Description Hide DescriptionThe solution of the onedimensional Schrödinger equation is discussed in the case of positiondependent mass. The general formalism is specified for potentials that are solvable in terms of generalized Laguerre polynomials and mass functions that are positive and bounded on the whole real axis. The resulting fourparameter potential is shown to belong to the class of “implicit” potentials. Closed expressions are obtained for the boundstate energies and the corresponding wave functions, including their normalization constants. The constant mass case is obtained by a specific choice of the parameters. It is shown that this potential contains both the harmonic oscillator and the Morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials. Possible generalizations of the method are outlined.

The symmetries of the fine gradings of associated with direct product of Pauli groups
View Description Hide DescriptionA grading of a Lie algebra is called fine if it could not be further refined. For a fine grading of a simple Lie algebra, we define its Weyl group to describe the symmetry of this grading. It is already known that the Weyl group of the fine grading of induced by the action of the group of the generalized Pauli matrices of rank is , where is the cyclic group of order . In this paper, we consider the fine grading of induced by the action of the group of fold tensor product of the generalized Pauli matrices of rank . We prove that its Weyl group is and is generated by transvections; therefore, this generalizes the previous result.

On the solution of the integral equation of scattering for finite range potentials
View Description Hide DescriptionIn the problem of the scattering of a particle in the presence of a finite range central potential, the integral equation for the partial wave is studied. By using a matrix method, the exact external solutions are expressed in terms of the Fredholm determinant, and the phase of turns out to be equal to the phase shift. As an example, an array of deltashell potentials is considered.

Twodimensional stationary Schrödinger equation via the dressing method: New exactly solvable potentials, wave functions, and their physical interpretation
View Description Hide DescriptionThe classes of exactly solvable multiline soliton potentials and corresponding wave functions of twodimensional stationary Schrödinger equation via dressing method are constructed and their physical interpretation is discussed.

Eigenkets of the deformed creation operator
View Description Hide DescriptionBy using the contour integral representation of function and the technique of integration within an ordered product of operators, we point out that the deformed creation operator possesses the eigenkets. A set of new completeness and orthogonality relations composed of the kets and bras which are not mutually Hermitian conjugates are derived. Application of the completeness relation in constructing the generalized Prepresentation of density operator is demonstrated.

Supersymmetry of the planar Dirac–Deser–Jackiw–Templeton system and of its nonrelativistic limit
View Description Hide DescriptionThe planar Dirac and the topologically massive vector gauge fields are unified into a supermultiplet involving no auxiliary fields. The superPoincaré symmetry emerges from the supersymmetry realized in terms of the deformed Heisenberg algebra underlying the construction. The nonrelativistic limit yields spin 1/2 as well as new, spin 1 “Lévy–Leblondtype” equations which, together, carry an superSchrödinger symmetry. Part of the latter has its origin in the universal enveloping algebra of the superPoincaré algebra.

Exact mapping between systemreservoir quantum models and semiinfinite discrete chains using orthogonal polynomials
View Description Hide DescriptionBy using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a onedimensional chain with only nearestneighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the shortrange interactions of the effective chain system permit these openquantum systems to be efficiently simulated by the density matrix renormalization group methods.
 Quantum Information and Computation

Nondisturbing quantum measurements
View Description Hide DescriptionWe consider pairs of discrete quantum observables (POVMs) and analyze the relation between the notions of nondisturbance, joint measurability, and commutativity. We specify conditions under which these properties coincide or differ—depending, for instance, on the interplay between the number of outcomes and the Hilbert space dimension or on algebraic properties of the effect operators. We also show that (non)disturbance is, in general, not a symmetric relation and that it can be decided and quantified by means of a semidefinite program.

Transformations of type entangled states
View Description Hide DescriptionThe transformations of type entangled states by using local operations assisted with classical communication are investigated. For this purpose, a parametrization of the type states which remains invariant under local unitary transformations is proposed and a complete characterization of the local operations carried out by a single party is given. These are used for deriving the necessary and sufficient conditions for deterministic transformations. A convenient upper bound for the maximum probability of distillation of arbitrary target states is also found.

Matrix permanent and quantum entanglement of permutation invariant states
View Description Hide DescriptionWe point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states. This connection allows us to interpret the permanent as an angle between vectors. By employing a recently introduced permanent inequality by Carlen et al. [Methods Appl. Anal.13, 1 (2006)], we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way.

Twosided bounds on minimumerror quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates
View Description Hide DescriptionIn a unified framework, we estimate the following quantities of interest in quantum information theory: (1) the minimumerror distinguishability of arbitrary ensembles of mixed quantum states; (2) the approximate reversibility of quantum dynamics in terms of entanglement fidelity (This is referred to as “channeladapted quantum error recovery” when applied to the composition of an encoding operation and a noise channel.); (3) the maximum overlap between a bipartite pure quantum state and a bipartite mixedstate that may be achieved by applying a local quantum operation to one part of the mixedstate; and (4) the conditional minentropy of bipartite quantum states. A refined version of the author’s techniques [J. Tyson, J. Math. Phys.50, 032016 (2009)] for bounding the first quantity is employed to give twosided estimates of the remaining three quantities. We obtain a closedform approximate reversal channel. Using a statedependent Kraus decomposition, our reversal may be interpreted as a quadratically weighted version of that of Barnum and Knill [J. Math. Phys.43, 2097 (2002)]. The relationship between our reversal and Barnum and Knill’s is therefore similar to the relationship between Holevo’s asymptotically optimal measurement [A. S. Kholevo, Theor. Probab. Appl.23, 411 (1978)] and the “pretty good” measurement of Belavkin [Stochastics1, 315 (1975)] and Hausladen and Wootters [J. Mod. Opt.41, 2385 (1994)]. In particular, we obtain relatively simple reversibility estimates without negative matrixpowers at no cost in tightness of our bounds. Our recovery operation is found to significantly outperform the socalled “transpose channel” in the simple case of depolarizing noise acting on half of a maximally entangled state. Furthermore, our overlap results allow the entangled input state and the output target state to differ, thus obtaining estimates in a somewhat more general setting. Using a result of König et al. [IEEE Trans. Inf. Theory55, 4337 (2009)], our maximum overlap estimate is used to bound the conditional minentropy of arbitrary bipartite states. Our primary tool is “small angle” initialization of an abstract generalization of the iterative schemes of Ježek et al. [Phys. Rev. A65, 060301 (2002)], Ježek et al. [Phys. Rev. A68, 012305 (2003)], and Reimpell and Werner [Phys. Rev. Lett.94, 080501 (2005)]. The monotonicity result of Reimpell [Ph.D. thesis, Technishe Universität, 2007] follows in greater generality.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

A commutant of system associated to the highest weight module of
View Description Hide DescriptionAnalogous to the commutant in the theory of associative algebras, one can construct a new subalgebra of a vertex algebra, known as the vertex algebra commutant. In this paper, for the highest weight module of Lie algebra, we describe a commutant of system by giving a finite set of generators and operator product expansion relations among generators.

Exceptional quantum subgroups for the rank two Lie algebras and
View Description Hide DescriptionExceptional modular invariants for the Lie algebras (at levels 2, 3, 7, and 12) and (at levels 3 and 4) can be obtained from conformal embeddings. We determine the associated algebras of quantum symmetries and discover or recover, as a byproduct, the graphs describing exceptional quantum subgroups of type or that encode their module structure over the associated fusion category. Global dimensions are given.

Integrated Lax formalism for principal chiral model
View Description Hide DescriptionBy solving the firstorder algebraic field equations which arise in the dual formulation of the principal chiral model (PCM), we construct an integrated Lax formalism built explicitly on the dual fields of the model rather than the currents. The Lagrangian of the dual scalar field theory is also constructed. Furthermore, we present the firstorder partial differential equation(PDE) system for an exponential parametrization of the solutions and discuss the integrability of this system.

Loop vertex expansion for theory in zero dimension
View Description Hide DescriptionIn this paper we extend the method of loop vertex expansion to interactions with degree higher than 4. As an example we provide through this expansion an explicit proof that the free energy of scalar theory in zero dimension is Borel–Le Roy summable of order . We detail the computations in the case of a interaction.
 General Relativity and Gravitation

On the relation between rigging inner product and master constraint direct integral decomposition
View Description Hide DescriptionCanonical quantization of constrained systems with firstclass constraints via Dirac’s operator constraint method proceeds by the theory of Rigged Hilbert spaces, sometimes also called refined algebraic quantization. This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the master constraint method has been proposed which replaces the individual constraints by a weighted sum of absolute squares of the constraints. Now the direct integral decomposition (DID) methods, which are closely related to Rigged Hilbert spaces, become available and have been successfully tested in various situations. It is relatively straightforward to relate the rigging inner product to the path integral that one obtains via reduced phase space methods. However, for the master constraint, this is not at all obvious. In this paper we find sufficient conditions under which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at least locally in phase space. Then the master constraint DID for those Abelian constraints can be directly related to the rigging map and therefore has a path integral formulation.

Ultrastatic spacetimes
View Description Hide DescriptionSeveral calculations in conformally staticspacetimes rely on the introduction of an ultrastatic background. I describe the general properties of ultrastatic spacetimes, and then focus on the problem of whether a given spacetime can be ultrastatic, or conformally ultrastatic, in more than one way. I show that the first possibility arises if and only if the spacetime contains regions that are products with a Minkowskian factor, and that the second arises if and only it contains regions whose spatial sections are conformal to a product space.
 Dynamical Systems

The Poisson geometry of SU(1,1)
View Description Hide DescriptionWe study the natural Poissonstructure on the group SU(1,1) and related questions. In particular, we give an explicit description of the Ginzburg–Weinstein isomorphism for the sets of admissible elements. We also establish an analog of Thompson’s conjecture for this group.