Volume 51, Issue 2, February 2010
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Ground state entanglement in onedimensional translationally invariant quantum systems
View Description Hide DescriptionWe examine whether it is possible for onedimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians for the infinite chain. The spectral gap of is . Moreover, for any state in the ground space of and any , there are regions of size with entanglemententropy. A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings [“An area law for one dimensional quantum systems,” J. Stat. Mech.: Theory Exp.2007 (08024)] gives a constant upper bound on the entanglemententropy for onedimensional ground states that is independent of the size of the region but exponentially dependent on , where is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglemententropy scales polynomially with . Previously, the best known such bound was logarithmic in .

Nonadditivity of Rényi entropy and Dvoretzky’s theorem
View Description Hide DescriptionThe goal of this note is to show that the analysis of the minimum output Rényi entropy of a typical quantum channel essentially amounts to applying Milman’s version of Dvoretzky’s theorem about almost Euclidean sections of highdimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden–Winter, disproving the additivity conjecture for the minimal output Rényi entropy (for ).

Quantum Zeno effect and dynamics
View Description Hide DescriptionIf frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finiterank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily bounded from below) generator as a generalized mean value Hamiltonian.

Selfadjoint Lyapunov variables, temporal ordering, and irreversible representations of Schrödinger evolution
View Description Hide DescriptionIn nonrelativistic quantum mechanics time enters as a parameter in the Schrödinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable—i.e., a selfadjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admits a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proven that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each into past and future subspaces.

Reformulating the Schrödinger equation as a Shabat–Zakharov system
View Description Hide DescriptionWe reformulate the secondorder Schrödinger equation as a set of two coupled firstorder differential equations, a socalled “Shabat–Zakharov system” (sometimes called a “Zakharov–Shabat” system). There is considerable flexibility in this approach, and we emphasize the utility of introducing an “auxiliary condition” or “gauge condition” that is used to cut down the degrees of freedom. Using this formalism, we derive the explicit (but formal) general solution to the Schrödinger equation. The general solution depends on three arbitrarily chosen functions, and a pathordered exponential matrix. If one considers path ordering to be an “elementary” process, then this represents complete quadrature, albeit formal, of the secondorder linear ordinary differential equation.

Any state solutions of the Hulthén potential in arbitrary dimensions
View Description Hide DescriptionThe energy spectra of the Hulthén effective potential in dimensions are obtained within the new quantization rule approach for any states. The interdimensional degeneracies among states are also presented. In the case of , the agreement between our results and those obtained by other methods is excellent.

Softcore Coulomb potentials and Heun’s differential equation
View Description Hide DescriptionSchrödinger’s equation with the attractive potential , , , , is shown, for general values of the parameters and , to be reducible to the confluent Heun equation in the case and to the generalized Heun equation in the case . In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases, this factor becomes a polynomial. The asymptotic iteration method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations for them. Detailed solutions for both cases are provided.

Exactly solvable nonseparable and nondiagonalizable twodimensional model with quadratic complex interaction
View Description Hide DescriptionWe study a quantum model with nonisotropic twodimensional oscillator potential but with additional quadratic interaction with imaginary coupling constant. It is shown that for a specific connection between coupling constant and oscillator frequencies, the modelis not amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is nondiagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of separation of variables in twodimensional systems is discussed.

Position dependent mass Schrödinger equation and isospectral potentials: Intertwining operator approach
View Description Hide DescriptionHere, we have studied first and secondorder intertwining approaches to generate isospectral partner potentials of position dependent (effective) mass Schrödinger equation. The secondorder intertwiner is constructed directly by taking it as secondorder linear differential operator with position dependent coefficients, and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained, which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to remove bound state(s), and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. It is shown that our results are consistent with the formulation of type A fold supersymmetry [T. Tanaka, J. Phys. A39, 219 (2006); A. GonzalezLopez and T. Tanaka, J. Phys. A39, 3715 (2006)] for the particular cases and , respectively.

A possible mathematics for the unification of quantum mechanics and general relativity
View Description Hide DescriptionThis paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces of functions of four spacetime variables , furnished with an additional indefinite inner product invariant under Poincaré transformations, and isomorphisms of these spaces that preserve the indefinite metric. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo transformations. Within these subspaces the usual quantum mechanics with Schrödinger evolution and as the evolution parameter is derived. Simultaneously, the Minkowski spacetime is isometrically embedded into , Poincaré transformations have unique extensions to isomorphisms of and the embedding commutes with Poincaré transformations. The main new result is a proof that the framework accommodates arbitrary pseudoRiemannian spacetimes furnished with the action of the diffeomorphism group.

Barycentric decomposition of quantum measurements in finite dimensions
View Description Hide DescriptionWe analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme points of the convex set are operator valued measures concentrated on a finite set of points of the outcome space, being the dimension of the Hilbert space. We prove that for secondcountable outcome spaces any POVM admits a Choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure. In the general case, Krein–Milman theorem is invoked to represent POVMs as barycenters of a certain set of POVMs concentrated on points of the outcome space.

A centerofmass principle for the multiparticle Schrödinger equation
View Description Hide DescriptionThe centerofmass principle is the key to the rapid computation of the interaction of a large number of classical particles. Electrons governed by the multiparticle Schrödinger equation have a much more complicated interaction mainly due to their spatial extent and the antisymmetry constraint on the total wave function of the combined electron system. We present a centerofmass principle for quantum particles that accounts for this spatial extent, the antisymmetry constraint, and the potential operators. We use it to construct an algorithm for computing a sizeconsistent approximate wave function for large systems with simple geometries.

The quadratic Fock functor
View Description Hide DescriptionWe construct the quadratic analog of the boson Fock functor. While in the first order (linear) case all contractions on the 1particle space can be second quantized, the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity. The encouraging fact is that it contains, as proper subgroups (i.e., the contractions), all the gauge transformations of second kind and all the a.e. invertible maps of into itself leaving the Lebesgue measure quasiinvariant (in particular, all diffeomorphism of ). This allows quadratic twodimensional quantization of gauge theories, of representations of the Witt group (in fact it continuous analog), of the Zamolodchikov hierarchy, and much more. Within this semigroup we characterize the unitary and the isometric elements and we single out a class of natural contractions.

On nonsingular potentials of Cox–Thompson inversion scheme
View Description Hide DescriptionWe establish a condition for obtaining nonsingular potentials using the Cox–Thompson inverse scattering method with one phase shift. The anomalous singularities of the potentials are avoided by maintaining unique solutions of the underlying Regge–Newton integral equation for the transformation kernel. As a byproduct, new inequality sequences of zeros of Bessel functions are discovered.
 Quantum Information and Computation

An application of decomposable maps in proving multiplicativity of low dimensional maps
View Description Hide DescriptionIn this paper, we present a class of maps for which the multiplicativity of the maximal output norm holds for and . The class includes all positive tracepreserving maps from to . In this sense, the result is a generalization of the corresponding result in the work of King and Koldan [“New multiplicativity results for qubit maps,” J. Math. Phys.47, 042106 (2006)], where the multiplicativity was proved for all positive tracepreserving maps from to with and . Interestingly, by contrast, the multiplicativity of norm was investigated in the context of quantum information theory and shown not to hold, in general, for high dimensional quantum channels [Hayden, P. and Winter, A., “Counterexamples to the maximal pnorm multiplicativity conjecture for all ,” Commun. Math. Phys.284, 263 (2008)]. Moreover, the Werner–Holevo channel, which is a map from to , is a counterexample for [Werner and Holevo, J. Math. Phys.43, 4353 (2002).].

Minimumerror state discrimination constrained by the nosignaling principle
View Description Hide DescriptionWe provide a bound on the minimum error when discriminating among quantum states, using the nosignaling principle. The bound is general in that it depends on neither dimensions nor specific structures of given quantum states to be discriminated among. We show that the bound is tight for the minimumerror state discrimination between symmetric (both pure and mixed) qubit states. Moreover, the bound can be applied to a set of quantum states for which the minimumerror state discrimination is not known yet. Finally, our results strengthen the quantitative connection between two nogo theorems, the nosignaling principle, and the no perfect state estimation.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

On superconformal characters and partition functions in three dimensions
View Description Hide DescriptionPossible short and semishort positive energy, unitary representations of the superconformal group in three dimensions are discussed. Corresponding character formulas are obtained, consistent with character formulas for the SO(3,2) conformal group, revealing long multiplet decomposition at unitarity bounds in a simple way. Limits, corresponding to reduction to various subalgebras, are taken in the characters that isolate contributions from fewer states, at a given unitarity threshold, leading to considerably simpler formula. Via these limits, applied to partition functions, closed formula for the generating functions for numbers of BPS operators in the free field limit of superconformal Chern–Simons theory and its supergravity dual are obtained in the large limit. Partial counting of semishort operators is performed and consistency between operator counting for the free field and supergravity limits with long multiplet decomposition rules is explicitly demonstrated. Partition functions counting certain protected scalar primary semishort operators, and their superconformal descendants, are proposed and computed for large . Certain chiral ring partition functions are discussed from a combinatorial perspective.

Cosmological particle creation in states of low energy
View Description Hide DescriptionThe recently proposed states of low energy provide a wellmotivated class of reference states for the quantized linear scalar field on cosmological Friedmann–Robertson–Walker spacetimes. The low energy property of a state is localized close to some value of the cosmological time coordinate. We present calculations of the relative cosmological particle production between a state of low energy at early time and another such state at later time. In an exponentially expanding Universe, we find that the particle production shows oscillations in the spatial frequency modes. The basis of the method for calculating the relative particle production is completely rigorous. Approximations are only used at the level of numerical calculation.

On the structure of Verma modules over the algebra
View Description Hide DescriptionIn this paper, we describe the structure of Verma modules over the algebra . We show that either a Verma module over is irreducible or its maximal submodule is cyclic.

Exact moduli space metrics for hyperbolic vortex polygons
View Description Hide DescriptionExact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic vortices are derived. These submanifolds, denoted as , are spaces of invariant vortex configurations with single vortices at the vertices of a regular polygon and coincident vortices at the polygon’s center. The geometric properties of are investigated, and it is found that is isometric to the hyperbolic plane of curvature . The geodesic flow on and a geometrically natural variant of geodesic flow recently proposed by Collie and Tong [“The dynamics of ChernSimons vortices,” Phys. Rev. D Part. Fields Gravit. Cosmol.78, 065013 (2008);eprint arXiv:hepth/0805.0602] are analyzed in detail.