Volume 51, Issue 7, July 2010
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

A deformation quantization theory for noncommutative quantum mechanics
View Description Hide DescriptionWe show that the deformation quantization of noncommutative quantum mechanics previously considered by Dias and Prata [“Weyl–Wigner formulation of noncommutative quantum mechanics,” J. Math. Phys.49, 072101 (2008)] and Bastos, Dias, and Prata [“Wigner measures in noncommutative quantum mechanics,” eprint arXiv:mathph/0907.4438v1; Commun. Math. Phys. (to appear)] can be expressed as a Weyl calculus on a double phase space. We study the properties of the starproduct thus defined and prove a spectral theorem for the stargenvalue equation using an extension of the methods recently initiated by de Gosson and Luef [“A new approach to the genvalue equation,” Lett. Math. Phys.85, 173–183 (2008)].

The geometric measure of multipartite entanglement and the singular values of a hypermatrix
View Description Hide DescriptionIt is shown that the geometric measure of entanglement of a pure multipartite state satisfies a polynomialequation, generalizing the singularvalue equation of the matrix of coefficients of a bipartite state. The equation is solved for a class of threequbit states.

Correlation of Dirac potentials and atomic inversion in cavity quantum electrodynamics
View Description Hide DescriptionControlling the time evolution of the population of two states in cavity quantum electrodynamics is necessary by tuning the modified Rabi frequency in which the extra classical effect of electromagnetic field is taken into account. The theoretical explanation underlying the perturbation of potential on spatial regime of bloch sphere is by the use of Bagrov–Baldiotti–Gitman–Shamshutdinova–Darboux transformations [Bagrov et al., “Darboux transformation for twolevel system,” Ann. Phys.14, 390 (2005)] on the electromagnetic field potential in onedimensional stationary Dirac model in which the Pauli matrices are the central parameters for controlling the collapse and revival of the Rabi oscillations. It is shown that by choosing in the transformation generates the parabolic potential causing the total collapse of oscillations, while yield the harmonic oscillator potentials ensuring the coherence of qubits.

Tree expansion in timedependent perturbation theory
View Description Hide DescriptionThe computational complexity of timedependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynmantype diagrams, etc.). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and nonperturbatively the convergence of Morita’s generalization of the Gell–Mann and Low wave function. We show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the RayleighSchrödinger timeindependent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree.

Maximal violation of Bell inequalities by position measurements
View Description Hide DescriptionWe show that it is possible to find maximal violations of the ClauserHorneShimonyHolt (CHSH) Bell inequality using only position measurements on a pair of entangled nonrelativistic free particles. The device settings required in the CHSH inequality are done by choosing one of two times at which position is measured. For different assignments of the “” outcome to positions, namely, to an interval, to a halfline, or to a periodic set, we determine violations of the inequalities and states where they are attained. These results have consequences for the hidden variable theories of Bohm and Nelson, in which the twotime correlations between distant particle trajectories have a joint distribution, and hence cannot violate any Bell inequality.

Orderdependent mappings: Strongcoupling behavior from weakcoupling expansions in nonHermitian theories
View Description Hide DescriptionA long time ago, it has been conjectured that a Hamiltonian with a potential of the form , real, has a real spectrum. This conjecture has been generalized to a class of the socalled symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an orderdependent mapping (ODM) in the whole complex plane of the coupling parameter , and that some information about the location of levelcrossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strongcoupling limit can be obtained from the initially weakcoupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of orderdependent “local” disk of convergence and analytic continuation by an ODM of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.

Spectral resolution of the Liouvillian of the Lindblad master equation for a harmonic oscillator
View Description Hide DescriptionA Lindblad master equation for a harmonic oscillator, which describes the dynamics of an open system, is formally solved. The solution yields the spectral resolution of the Liouvillian, that is, all eigenvalues and eigenprojections are obtained. This spectral resolution is discussed in depth in the context of the biorthogonal system and the rigged Hilbert space, and the contribution of each eigenprojection to expectation values of physical quantities is revealed. We also construct the ladder operators of the Liouvillian, which clarify the structure of the spectral resolution.

Whittaker–Hill equation and semifinitegap Schrödinger operators
View Description Hide DescriptionA periodic onedimensional Schrödinger operator is called semifinite gap if every second gap in its spectrum is eventually closed. We construct explicit examples of semifinitegap Schrödinger operators in trigonometric functions by applying Darboux transformations to the Whittaker–Hill equation. We give a criterion for the regularity of the corresponding potentials and investigate the spectral properties of the new operators.
 Quantum Information and Computation

Conjugate degradability and the quantum capacity of cloning channels
View Description Hide DescriptionA quantum channel is conjugate degradable if the channel’s environment can be simulated up to complex conjugation using the channel’s output. For all such channels, the quantum capacity can be evaluated using a singleletter formula. In this article we introduce conjugate degradability and establish a number of its basic properties. We then use it to calculate the quantum capacity of to and 1 to universal quantum cloning machines as well as the quantum capacity of a channel that arises naturally when data are being transmitted to an accelerating receiver. All the channels considered turn out to have strictly positive quantum capacity, meaning they could be used as part of a communication system to send quantum states reliably.

Hadamard matrices from mutually unbiased bases
View Description Hide DescriptionAn analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the case that leads to a rich family of eightdimensional Hadamard matrices that depend on five arbitrary parameters whose modulus is equal to unity.

Exponential error rates in multiple state discrimination on a quantum spin chain
View Description Hide DescriptionWe consider decision problems on finite sets of hypotheses represented by pairwise different shiftinvariant states on a quantum spin chain. The decision in favor of one of the hypotheses is based on outcomes of generalized measurements performed on local states on blocks of finite size. We assume the existence of the mean quantum Chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum Chernoff distance. We establish that this minimum specifies an asymptotic bound on the exponential rate of decay of the averaged probability of rejecting the true state in increasing block size, if the mean quantum Chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem. This assumption is, in particular, fulfilled by shiftinvariant product states (independent and identically distributed states). Further, we provide a constructive proof for the existence of a sequence of quantum tests in increasing block length with an error exponent which equals, up to a factor, the mean generalized quantum Chernoff distance. The factor depends on the configuration of the hypothetic states with respect to the binary quantum Chernoff distances. It can be arbitrary close to 1 and is never less than for being the number of different pairs of states.

Householder factorizations of unitary matrices
View Description Hide DescriptionA method to construct all representations of finite dimensional unitary matrices as the product of Householder reflections is given. By arbitrarily severing the state space into orthogonal subspaces, the method may, e.g., identify the entangling and singlecomponent quantum operations that are required in the engineering of quantum states of composite (multipartite) systems. Earlier constructions are shown to be extreme cases of the unifying scheme that is presented here.

Matrix pencils and entanglement classification
View Description Hide DescriptionQuantum entanglement plays a central role in quantum information processing. A main objective of the theory is to classify different types of entanglement according to their interconvertibility through manipulations that do not require additional entanglement to perform. While bipartite entanglement is well understood in this framework, the classification of entanglements among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension . Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the wellstudied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomialtime algorithm for deciding when two states are SLOCC equivalent. We then proceed to present a canonical form for all states based on the matrix pencil construction such that two states are equivalent if and only if they have the same canonical form. Besides recovering the previously known 26 distinct SLOCC equivalence classes in systems, we also determine the hierarchy between these classes.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

The causal perturbation expansion revisited: Rescaling the interacting Dirac sea
View Description Hide DescriptionThe causal perturbation expansion defines the Dirac sea in the presence of a timedependent external field. It yields an operator whose image generalizes the vacuum solutions of negative energy and thus gives a canonical splitting of the solution space into two subspaces. After giving a selfcontained introduction to the ideas and techniques, we show that this operator is, in general, not idempotent. We modify the standard construction by a rescaling procedure giving a projector on the generalized negativeenergy subspace. The resulting rescaled causal perturbation expansion uniquely defines the fermionic projector in terms of a series of distributional solutions of the Dirac equation. The technical core of the paper is to work out the combinatorics of the expansion in detail. It is also shown that the fermionic projector with interaction can be obtained from the free projector by a unitary transformation. We finally analyze the consequences of the rescaling procedure on the lightcone expansion.

Yang–Mills equations of motion for the Higgs sector of SU(3)equivariant quiver gauge theories
View Description Hide DescriptionWe consider SU(3)equivariant dimensional reduction of Yang–Mills theory on spaces of the form , with equals either or . For the corresponding quiver gauge theory, we derive the equations of motion and construct some specific solutions for the Higgs fields using different gauge groups. Specifically, we choose the gauge groups U(6) and U(8) for the space , as well as the gauge group U(3) for the space , and derive Yang–Mills equations for the latter one using a spin connection endowed with a nonvanishing torsion. We find that a specific value for the torsion is necessary in order to obtain nontrivial solutions of Yang–Mills equations. Finally, we take the space and derive the equations of motion for the Higgs sector for the gauge theory.
 General Relativity and Gravitation

Linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold
View Description Hide DescriptionWe study the linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold. In particular, we prove that these equations are linearization stable in the neighborhood of vacuum solutions for a nonpositive cosmological constant and of Friedman–Lemaître–Robertson–Walker spaces in a certain range of decays. We also prove that this result is no longer true for faster decays. The construction of the counterexamples is based on a new construction of transverse traceless tensors on the Euclidean space and on positive energy theorems.

Mixed potentials in radiative stellar collapse
View Description Hide DescriptionWe study the behavior of a radiating star when the interior expanding, shearing fluid particles are traveling in geodesic motion. We demonstrate that it is possible to obtain new classes of exact solutions in terms of elementary functions without assuming a separable form for the gravitational potentials or initially fixing the temporal evolution of the model unlike earlier treatments. A systematic approach enables us to write the junction condition as a Riccati equation which under particular conditions may be transformed into a separable equation. New classes of solutions are generated which allow for mixed spatial and temporal dependence in the metric functions. We regain particular models found previously from our general classes of solutions.

de Sitter breaking through infrared divergences
View Description Hide DescriptionJust because the propagator of some field obeys a de Sitter invariant equation does not mean it possesses a de Sitter invariant solution. The classic example is the propagator of a massless, minimally coupled scalar. We show that the same thing happens for massive scalars with and for massive transverse vectors with , where is the dimension of spacetime and is the Hubble parameter. Although all masses in these ranges give infrared divergent mode sums, using dimensional regularization (or any other analytic continuation technique) to define the mode sums leads to the incorrect conclusion that de Sitter invariant solutions exist except at discrete values of the masses.

The Hamiltonian formulation for the dynamics of a multishell selfgravitating system
View Description Hide DescriptionHamiltonian function describing a system composed of gravitating shells in general relativity is derived from general considerations and its dynamics is presented. The results appear to be promising for the description of colliding system of astrophysical and cosmological interest.

Cylindrically symmetric vacuum solutions in higher dimensional Brans–Dicke theory
View Description Hide DescriptionHigher dimensional, static, cylindrically symmetric vacuum solutions with and without a cosmological constant in the Brans–Dicke theory are presented. We show that for a negative cosmological constant and for specific values of the parameters, a particular subclass of these solutions includes higher dimensional topological black holetype solutions with a flat horizon topology. We briefly extend our discussion to stationary vacuum and vacuum solutions.