Volume 51, Issue 9, September 2010
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
51(2010); http://dx.doi.org/10.1063/1.3481567View Description Hide Description
States 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.
51(2010); http://dx.doi.org/10.1063/1.3491333View Description Hide Description
We argue that the continuous-time 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.
51(2010); http://dx.doi.org/10.1063/1.3483716View Description Hide Description
The solution of the one-dimensional Schrödinger equation is discussed in the case of position-dependent 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 four-parameter potential is shown to belong to the class of “implicit” potentials. Closed expressions are obtained for the bound-state 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.
51(2010); http://dx.doi.org/10.1063/1.3492830View Description Hide Description
A 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.
51(2010); http://dx.doi.org/10.1063/1.3483692View Description Hide Description
In 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 delta-shell potentials is considered.
51(2010); http://dx.doi.org/10.1063/1.3485041View Description Hide Description
By 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 P-representation of density operator is demonstrated.
51(2010); http://dx.doi.org/10.1063/1.3478558View Description Hide Description
The planar Dirac and the topologically massive vector gauge fields are unified into a supermultiplet involving no auxiliary fields. The super-Poincaré 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–Leblond-type” equations which, together, carry an super-Schrödinger symmetry. Part of the latter has its origin in the universal enveloping algebra of the super-Poincaré algebra.
Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials51(2010); http://dx.doi.org/10.1063/1.3490188View Description Hide Description
By 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 one-dimensional chain with only nearest-neighbor 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 short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.
- Quantum Information and Computation
51(2010); http://dx.doi.org/10.1063/1.3480658View Description Hide Description
We 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.
51(2010); http://dx.doi.org/10.1063/1.3481573View Description Hide Description
The 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.
51(2010); http://dx.doi.org/10.1063/1.3464263View Description Hide Description
We 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.
Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates51(2010); http://dx.doi.org/10.1063/1.3463451View Description Hide Description
In a unified framework, we estimate the following quantities of interest in quantum information theory: (1) the minimum-error 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 “channel-adapted 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 mixed-state that may be achieved by applying a local quantum operation to one part of the mixed-state; and (4) the conditional min-entropy 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 two-sided estimates of the remaining three quantities. We obtain a closed-form approximate reversal channel. Using a state-dependent 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 matrix-powers at no cost in tightness of our bounds. Our recovery operation is found to significantly outperform the so-called “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 min-entropy 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)
51(2010); http://dx.doi.org/10.1063/1.3476318View Description Hide Description
Analogous 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.
51(2010); http://dx.doi.org/10.1063/1.3476319View Description Hide Description
Exceptional 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 by-product, the graphs describing exceptional quantum subgroups of type or that encode their module structure over the associated fusion category. Global dimensions are given.
51(2010); http://dx.doi.org/10.1063/1.3480667View Description Hide Description
By solving the first-order 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 first-order partial differential equation(PDE) system for an exponential parametrization of the solutions and discuss the integrability of this system.
51(2010); http://dx.doi.org/10.1063/1.3460320View Description Hide Description
In 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
51(2010); http://dx.doi.org/10.1063/1.3486359View Description Hide Description
Canonical quantization of constrained systems with first-class 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.
51(2010); http://dx.doi.org/10.1063/1.3485599View Description Hide Description
Several calculations in conformally staticspace-times rely on the introduction of an ultrastatic background. I describe the general properties of ultrastatic space-times, and then focus on the problem of whether a given space-time can be ultrastatic, or conformally ultrastatic, in more than one way. I show that the first possibility arises if and only if the space-time 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
51(2010); http://dx.doi.org/10.1063/1.3461878View Description Hide Description
We 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.