Showing papers in "Journal of Mathematical Physics in 2013"

On quantum Rényi entropies: A new generalization and some properties

Abstract: The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.

Ground state solutions for nonlinear fractional Schrödinger equations in RN

Abstract: We construct solutions to a class of Schrodinger equations involving the fractional Laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.

Monotonicity of a relative Rényi entropy

Abstract: We show that a recent definition of relative Renyi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Muller-Lennert et al. [“On quantum Renyi entropies: A new definition, some properties,” J. Math. Phys. 54, 122203 (2013); e-print arXiv:1306.3142v1; see also e-print arXiv:1306.3142].

Sandwiched Rényi divergence satisfies data processing inequality

Abstract: Sandwiched (quantum) α-Renyi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglement-breaking channels,” preprint arXiv:1306.1586 (2013)] and Muller-Lennert et al. [“On quantum Renyi entropies: a new definition, some properties and several conjectures,” preprint arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular, we show that sandwiched α-Renyi divergence satisfies the data processing inequality for all values of α > 1. Moreover we prove that α-Holevo information, a variant of Holevo information defined in terms of sandwiched α-Renyi divergence, is super-additive. Our results are based on Holder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.

Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms

Abstract: In recent three-loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short S-sums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincare iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the S-sums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation with respect to the external summation index and different multi-argument relations, for the compactification of S-sum expressions. Finally, we calculate algebraic relations for infinite S-sums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package HarmonicSums.

The two-loop sunrise graph with arbitrary masses

Abstract: We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve.

Quantum logarithmic Sobolev inequalities and rapid mixing

Abstract: A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative Lp-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed. This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants. Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained, incl...

Fractional Schrödinger equation for a particle moving in a potential well

Abstract: In this paper, the fractional Schrodinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrodinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrodinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrodinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power ...

Flag-dipole spinor fields in ESK gravities

Abstract: We consider the Riemann-Cartan geometry as a basis for the Einstein-Sciama-Kibble theory coupled to spinor fields: we focus on f(R) and conformal gravities, regarding the flag-dipole spinor fields, type-(4) spinor fields under the Lounesto classification. We study such theories in specific cases given, for instance, by cosmological scenarios: we find that in such background the Dirac equation admits solutions that are not Dirac spinor fields, but in fact the aforementioned flag-dipoles ones. These solutions are important from a theoretical perspective, as they evince that spinor fields are not necessarily determined by their dynamics, but also a discussion on their structural (algebraic) properties must be carried off. Furthermore, the phenomenological point of view is shown to be also relevant, since for isotropic Universes they circumvent the question whether spinor fields do undergo the Cosmological Principle.

Note on intrinsic decay rates for abstract wave equations with memory

Abstract: In this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g′ + H(g) ⩽ 0, s ⩾ 0 where H(s) is a given continuous, positive, increasing, and convex function such that H(0) = 0. We shall develop an intrinsic method, based on the main idea introduced by Lasiecka and Tataru [“Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,” Differential and Integral Equations 6, 507–533 (1993)], for determining decay rates of the energy given in terms of the function H(s). This will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ODE. We shall show that the obtained result, while generalizing previous results obtained in the literature, is also capable of proving optimal decay rates for polynomially decaying memory kernels (H(s) ∼ sp) and for the full range of admissible parameters p ∈ [1, 2). While such result has been known for certain restrictive ranges of the parameters p ∈ [1, 3/2...

A hierarchy of topological tensor network states

Abstract: We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.

Towards an invariant geometry of double field theory

Abstract: We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an “index-free” proof of the algebraic Bianchi identity. Finally, we analyze to what extent the generalized Riemann tensor encodes the curvatures of Riemannian geometry. We show that it contains the conventional Ricci tensor and scalar curvature but not the full Riemann tensor, suggesting the possibility of a further extension of this framework.

Fractional wave equation and damped waves

Abstract: In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of the same order α, 1 ⩽ α ⩽ 2, both in space and in time. We show that this feature is a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum of its fundamental solution and its gravity and “mass” centers. Moreover, the first, the second, and the Smith centrovelocities of the damped waves described by the fractional wave equation are constant and depend just on the equation order α. The fundamental solution of the fractional wave equation is determined and shown to be a spatial probability density function evolving in time all whose moments of order less than α are finite. To illustrate analytical findings, results of numerical calculations and plots are presented.

Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups

Abstract: The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a generalized notion of (non-commutative) Fourier transform, different from standard harmonic analysis, has been recently developed, and found several applications, especially in the quantum gravity literature. We show that this algebra representation can be defined on the sole basis of a quantization map of the classical Poisson algebra, and identify the conditions for its existence. In particular, the corresponding non-commutative star-product carried by this representation is obtained directly from the quantization map via deformation quantization. We then clarify under which conditions a unitary intertwiner between such algebra representation and the usual group representation can be constructed giving rise to the non-commutative plane waves and consequently, the non-commutative Fourier transform. The compact groups U(1) and SU(2) are considered for different choices of quantization maps, such as the symmetric and the Duflo map, and we exhibit the corresponding star-products, algebra representations, and non-commutative plane waves.

On a solution of the Schrödinger equation with a hyperbolic double-well potential

Abstract: We report a solution of the one-dimensional Schrodinger equation with a hyperbolic double-well confining potential via a transformation to the so-called confluent Heun equation. We discuss the requirements on the parameters of the system in which a reduction to confluent Heun polynomials is possible, representing the wavefunctions of bound states.

Ground state solutions of asymptotically linear fractional Schrödinger equations

Abstract: This paper is devoted to a time-independent fractional Schrodinger equation of the form (−Δ)su+V(x)u=f(x,u)inRN, where N ⩾ 2, s ∈ (0, 1), (−Δ)s stands for the fractional Laplacian. We apply the variational methods to obtain the existence of ground state solutions when f(x, u) is asymptotically linear with respect to u at infinity.

Soliton solutions for quasilinear Schrödinger equations

Abstract: It is established the existence of nontrivial solutions for quasilinear Schrodinger equations with subcritical or critical exponents, which appear from plasma physics as well as high-power ultrashort laser in matter

Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables

Abstract: The antiperiodic transfer matrices associated to higher spin representations of the rational 6-vertex Yang-Baxter algebra are analyzed by generalizing the approach introduced recently in the framework of Sklyanin's quantum separation of variables (SOV) for cyclic representations, spin-1/2 highest weight representations, and also for spin-1/2 representations of the 6-vertex reflection algebra. Such SOV approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on Bethe ansatz and Baxter Q-operator. In particular, we both prove the completeness of the SOV characterization of the transfer matrix spectrum and its simplicity. Then, the derived characterization of local operators by Sklyanin's quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products.

New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials

Abstract: In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.

On fractional Schrödinger equation in RN with critical growth

Abstract: In this paper, we study the following nonlinear fractional Schrodinger equation with critical exponent h2α(−Δ)αu+V(x)u=|u|2α*−2u+λ|u|q−2u,x∈RN, where h is a small positive parameter, 0 2α, λ > 0 is a parameter. The potential V:RN→R is a positive continuous function satisfying some natural assumptions. By using variational methods, we obtain the existence of solutions in the following case: if 2 0 such that for all λ ⩾ λ0, we show that it has one nontrivial solution and there exist at least catΛδ(Λ) nontrivial solutions; if max{2,4αN−2α} 0.

On the solvability of the quantum Rabi model and its 2-photon and two-mode generalizations

Abstract: We study the solvability of the time-independent matrix Schrodinger differential equations of the quantum Rabi model and its 2-photon and two-mode generalizations in Bargmann Hilbert spaces of entire functions. We show that the Rabi model and its 2-photon and two-mode analogs are quasi-exactly solvable. We derive the exact, closed-form expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces. Up to a normalization factor, the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations.

Towards a double field theory on para-Hermitian manifolds

Abstract: In a previous paper, we have shown that the geometry of double field theory has a natural interpretation on flat para-Kahler manifolds. In this paper, we show that the same geometric constructions can be made on any para-Hermitian manifold. The field is interpreted as a compatible (pseudo-)Riemannian metric. The tangent bundle of the manifold has a natural, metric-compatible bracket that extends the C-bracket of double field theory. In the para-Kahler case, this bracket is equal to the sum of the Courant brackets of the two Lagrangian foliations of the manifold. Then, we define a canonical connection and an action of the field that correspond to similar objects of double field theory. Another section is devoted to the Marsden-Weinstein reduction in double field theory on para-Hermitian manifolds. Finally, we give examples of fields on some well-known para-Hermitian manifolds.

Local deformed semicircle law and complete delocalization for Wigner matrices with random potential

Abstract: We consider Hermitian random matrices of the form H = W + λV, where W is a Wigner matrix and V is a diagonal random matrix independent of W. We assume subexponential decay for the matrix entries of W and we choose λ ∼ 1 so that the eigenvalues of W and λV are of the same order in the bulk of the spectrum. In this paper, we prove for a large class of diagonal matrices V that the local deformed semicircle law holds for H, which is an analogous result to the local semicircle law for Wigner matrices. We also prove complete delocalization of eigenvectors and other results about the positions of eigenvalues.

Random loop representations for quantum spin systems

Abstract: We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 12 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)-invariance. Quantum spin correlations are given by loop correlations. Decay of correlations is proved in 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase.

Some classes of renormalizable tensor models

Abstract: We identify new families of renormalizable tensor models from anterior renormalizable tensor models via a mapping capable of reducing or increasing the rank of the theory without having an effect on the renormalizability property. Mainly, a version of the rank 3 tensor model as defined by Ben Geloun and Samary [Ann. Henri Poincare 14, 1599 (2013); e-print arXiv:1201.0176 [hep-th]]10.1007/s00023-012-0225-5 and the Grosse-Wulkenhaar model in 4D and 2D generate three different classes of renormalizable models. The proof of the renormalizability is fully performed for the first reduced model. The same procedure can be applied for the remaining cases. Interestingly, we find that, due to the peculiar behavior of anisotropic wave function renormalizations, the rank 3 tensor model reduced to a matrix model generates a simple super-renormalizable vector model.

Rapid mixing implies exponential decay of correlations

Abstract: We provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the Log-Sobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information—a much more stringent form of correlation decay. As a consequence, in the latter case we get an area law (with logarithmic corrections) for the mutual information. As a further corollary, we obtain a stability theorem for local distant perturbations. We also demonstrate that gapped free-fermionic systems exhibit clustering of correlations in the covariance and in the mutual information. We conclude with a discussion of the implications of these results for the classical simulation of open quantum systems with matrix-product operators and the robust dissipative preparation of topologically ordered states of lattice spin systems.

Multidimensional Yamada-Watanabe theorem and its applications to particle systems

Abstract: We prove a multidimensional version of the Yamada-Watanabe theorem, i.e., a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions. It implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrix-valued stochastic processes, called a “spectral” matrix Yamada-Watanabe theorem. The multidimensional Yamada-Watanabe theorem is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes, Wishart and Jacobi matrix processes. The β-versions of these particle systems are also considered.

Constraint analysis for variational discrete systems

Abstract: A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves and naturally incorporates both constant and evolving phase spaces, the latter of which is necessary for a time varying discretization. The different roles of constraints in the discrete and the conditions under which they are first or second class and/or symmetry generators are clarified. The (non-) preservation of constraints and the symplectic structure is discussed; on evolving phase spaces the number of constraints at a fixed time step depends on the initial and final time step of evolution. Moreover, the definition of observables and a reduced phase space is provided; again, on evolving phase spaces the notion of an observable as a propagating degree of freedom requires specification of an initial and final step and crucially depends on this choice, in contrast to the continuum. However, upon restriction to translation invariant systems, one regains the usual time step independence of canonical concepts. This analysis applies, e.g., to discrete mechanics, lattice field theory, quantum gravity models, and numerical analysis.

On the regularity for the Boussinesq equations in a bounded domain

Abstract: We address the regularity for the 2D Boussinesq equations with zero diffusivity and positive viscosity on an open bounded smooth domain Ω. In particular, we prove that for the initial velocity in H2(Ω) and the initial density in H1(Ω) there exists a global in time solution for which the velocity and the density are locally in time bounded functions in H2 and H1, respectively.

On Twistors and Conformal Field Theories from Six Dimensions

Abstract: We discuss chiral zero-rest-mass field equations on six-dimensional space-time from a twistorial point of view. Specifically, we present a detailed cohomological analysis, develop both Penrose and Penrose–Ward transforms, and analyse the corresponding contour integral formulae. We also give twistor space action principles. We then dimensionally reduce the twistor space of six-dimensional space-time to obtain twistor formulations of various theories in lower dimensions. Besides well-known twistor spaces, we also find a novel twistor space amongst these reductions, which turns out to be suitable for a twistorial description of self-dual strings. For these reduced twistor spaces, we explain the Penrose and Penrose–Ward transforms as well as contour integral formulae.