Table of contents

We address the non-Gaussianity (nG) of states obtained by weakly perturbing a Gaussian state and investigate the relationships with quantum estimation. For classical perturbations, i.e. perturbations to eigenvalues, we found that the nG of the perturbed state may be written as the quantum Fisher information (QFI) distance minus a term depending on the infinitesimal energy change, i.e. it provides a lower bound to statistical distinguishability. Upon moving on isoenergetic surfaces in a neighbourhood of a Gaussian state, nG thus coincides with a proper distance in the Hilbert space and exactly quantifies the statistical distinguishability of the perturbations. On the other hand, for perturbations leaving the covariance matrix unperturbed, we show that nG provides an upper bound to the QFI. Our results show that the geometry of non-Gaussian states in the neighbourhood of a Gaussian state is definitely not trivial and cannot be subsumed by a differential structure. Nevertheless, the analysis of perturbations to a Gaussian state reveals that nG may be a resource for quantum estimation. The nG of specific families of perturbed Gaussian states is analysed in some detail with the aim of finding the maximally non-Gaussian state obtainable from a given Gaussian one.

Recently, simulating the statistics of the singlet state with non-quantum resources has generated much interest. The singlet state statistics can be simulated by 1 bit of classical communication without using any further non-local correlation. But, interestingly, the singlet state statistics can also be simulated with no classical cost if a non-local box is used. In the first case, the output is completely deterministic whereas in the second case, outputs are completely random. We suggest a (possibly) signaling correlation resource which successfully simulates the singlet statistics, and subsequently leads to a complementary relation between the required classical bits and randomness in the local output involved in the simulation. Our result reproduces the above two models of simulation as extreme cases. We also discuss some important features in Leggett's non-local model and the model presented by Groblacher et al.

We study the dynamics of a lazy random walker who is inactive for extended times and tries to make up for her laziness with very large jumps. She remains in a condition of rest for a time τ derived from a waiting-time distribution , with μ_W < 2, thereby making jumps only from time to time from a position x to a position x' of a one-dimensional path. However, when the random walker jumps, she moves by quantities l = |x − x'| derived randomly from a distribution , with μ_ξ > 1. The most convenient choice to make up for the random walker laziness would be to select μ_ξ < 3, which in the ordinary case μ_W > 2 would produce Lévy flights with scaling δ = 1/(μ_ξ − 1) and consequently super-diffusion. According to the Sparre Andersen theorem, the distribution density of the first times to go from x_A to x_B > x_A has the inverse power law form with μ_FPT = μ_SA = 1.5. We find the surprising result that there exists a region of the phase space (μ_ξ, μ_W) with μ_W < 2, where μ_FPT > μ_SA and the lazy walker compensates for her laziness. There also exists an extended region breaking the Sparre Andersen theorem, where the lazy runner cannot compensate for her laziness. We make conjectures concerning the possible relevance of this mathematical prediction, supported by numerical calculations, for the problem of animal foraging.

We present various Miura-type transformations that exist between integrable lattice equations, which lead to some new and quite unexpected relations between these lattice equations. In particular, we show that in the discrete case, contrary to the continuous one, the sine-Gordon and mKdV equations are essentially the same. We also examine two new equations recently proposed by Hydon and Viallet and show that they can be transformed to the discrete mKdV and/or sine-Gordon equations.

We investigate the critical behaviour of a probabilistic mixture of cellular automata (CA) rules 182 and 200 (in Wolfram's enumeration scheme) by mean-field analysis and Monte Carlo simulations. We found that as we switch off one CA and switch on the other by the variation of the single parameter of the model, the probabilistic CA (PCA) goes through an extinction–survival-type phase transition, and the numerical data indicate that it belongs to the directed percolation universality class of critical behaviour. The PCA displays a characteristic stationary density profile and a slow, diffusive dynamics close to the pure CA 200 point that we discuss briefly. Remarks on an interesting related stochastic lattice gas are addressed in the conclusions.

The directed transport of Brownian particles requires a system with an asymmetry and with non-equilibrium noise. Here we investigate numerically alternative ways of fulfilling these requirements for a two-state Brownian motor, realized with Brownian particles alternating between two phase-shifted, symmetric potentials. We show that, besides the previously known spatio-temporal asymmetry based on unequal transfer rates between the potentials, inequalities in the potential depths, the frictions, or the equilibrium temperatures of the two potentials also generate the required asymmetry. We also show that the effects of the thermal noise and the noise of the transfer's randomness depend on the way the asymmetry is induced.

We show that the general heavenly equation, suggested recently by Doubrov and Ferapontov (2010 arXiv:0910.3407v2 [math.DG]), governs anti-self-dual (ASD) gravity. We derive ASD Ricci-flat vacuum metric governed by the general heavenly equation, null tetrad and basis of 1-forms for this metric. We present algebraic exact solutions of the general heavenly equation as a set of zeros of homogeneous polynomials in independent and dependent variables. A real solution is obtained for the case of a neutral signature.

In this paper, we investigate the three-dimensional Schrödinger operator with a periodic, relative to a lattice Ω of potential q. A special class V of the periodic potentials is constructed, which is easily and constructively determined from the spectral invariants. First, we give an algorithm for the unique determination of the potential q ∊ V of the three-dimensional Schrödinger operator from the spectral invariants that were determined constructively from the given Bloch eigenvalues. Then, we consider the stability of the algorithm with respect to the spectral invariants and Bloch eigenvalues. Finally, we prove that there are no other periodic potentials in the set of large class of functions whose Bloch eigenvalues coincides with the Bloch eigenvalues of q ∊ V.

By means of the similarity transformation, we obtain exact self-similar solutions (similaritons) of the generalized cubic-quintic (CQ) nonlinear Schrödinger equation with spatially inhomogeneous group velocity dispersion, CQ nonlinearity and amplification or attenuation. Exact balance conditions between the dispersion, nonlinearity and the gain/loss have been obtained. As an example, we investigate their propagation dynamics in the dispersion decreasing fiber (DDF). Considering the fluctuation of the fiber parameter in real application, the exact balance conditions do not satisfy, and so we perform direct numerical analysis with initial 10% white noise for the bright similariton in both the DDF and the periodic distributed amplification system. Numerical calculations indicate stable propagation of the bright similariton over tens of dispersion lengths. These ultrashort self-similar optical waves are potentially useful for all-optical data-processing schemes and the design of beam compressors and amplifiers.

This paper is concerned with the study of Poisson integrators. We are interested in Lie–Poisson systems on . First, we focus on Poisson integrators for constant Poisson systems and the transformations used for transforming Lie–Poisson structures to constant Poisson structures. Then, we construct local Poisson integrators for Lie–Poisson systems on . Finally, we present the results of numerical experiments for two Lie–Poisson systems and compare our Poisson integrators with other known methods.

We investigate solutions of a non-relativistic wave equation in hyperspherical coordinates for a diatomic molecule system interacting with a generalized Kratzer potential. Rovibrational eigenvalues and corresponding wavefunctions of non-relativistic diatomic molecules have been determined within the framework of the asymptotic iteration method. Certain fundamental conditions for the applications of the asymptotic iteration method, such as a suitable asymptotic form for the wave-function and the termination condition for the iteration process, are discussed. N-dimensional bound state eigenfunction solutions used in studying the dynamical variables of diatomic molecules are obtained in terms of a confluent hypergeometric function and a generalized Laguerre polynomial. This systematic approach is tested by calculating the rovibrational energy spectra of hydrogen and sodium chloride molecules.

We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order. In numerical experiments we compare our integrators with some other numerical schemes, including the standard discrete gradient method, the leap-frog scheme and a symplectic scheme of fourth order. We study the error accumulation for a very long time and the conservation of the energy integral.

Starting from basic identities of the group E₈, we perform progressive reductions, namely decompositions with respect to the maximal and symmetric embeddings of E₇ × SU(2) and then of E₆ × U(1). This procedure provides a systematic approach to the basic identities involving invariant primitive tensor structures of various irreps of finite-dimensional exceptional Lie groups. We derive novel identities for E₇ and E₆, highlighting the E₈ origin of some well-known ones. In order to elucidate the connections of this formalism to four-dimensional Maxwell–Einstein supergravity theories based on symmetric scalar manifolds (and related to irreducible Euclidean Jordan algebras, the unique exception being the triality-symmetric stu model), we then derive a fundamental identity involving the unique rank-4 symmetric invariant tensor of the 0-brane charge symplectic irrep of U-duality groups, with potential applications in the quantization of the charge orbits of supergravity theories, as well as in the study of multi-center black hole solutions therein.

Topological subsystem codes proposed recently by Bombin are quantum error-correcting codes defined on a two-dimensional grid of qubits that permit reliable quantum information storage with a constant error threshold. These codes require only the measurement of two-qubit nearest-neighbor operators for error correction. In this paper, we demonstrate that topological subsystem codes (TSCs) can be viewed as generalizations of Kitaev's honeycomb model to 3-valent hypergraphs. This new connection provides a systematic way of constructing TSCs and analyzing their properties. We also derive a necessary and sufficient condition under which a syndrome measurement in a subsystem code can be reduced to measurements of the gauge group generators. Furthermore, we propose and implement some candidate decoding algorithms for one particular TSC assuming perfect error correction. Our Monte Carlo simulations indicate that this code, which we call the five-square code, has a threshold against depolarizing noise of at least 2%.

An analytical method is proposed to implement any classical Boolean function in a small quantum system by taking the advantage of its electronic transport properties. The logical input, α = {α₁, ..., α_N}, is used to control well-identified parameters of the Hamiltonian of the system noted . The logical output is encoded in the tunneling current intensity passing through the quantum system when connected to conducting electrodes. It is demonstrated how to implement the six symmetric two-input/one-output Boolean functions in a quantum system. This system can be switched from one logic function to another by changing its structural parameters. The stability of the logic gates is discussed, perturbing the Hamiltonian with noise sources and studying the effect of decoherence.

We consider an open bipartite quantum system with dissipative dynamics generated by , where are generators of Lindblad type and 0 < ε ≪ 1. In order to study the entanglement of the stationary states of , we develop a perturbative approach and apply it to the physically significant case when generates a reversible unitary dynamics, while is a purely dissipative perturbation.

For any even n qubits, we establish four stochastic local operations and classical communication (SLOCC) equations and construct four SLOCC polynomials (not complete) of degree 2^n/2, which can be exploited for the SLOCC classification (not complete) of any even n qubits. In light of the SLOCC equations, we propose several different genuine entangled states of even n qubits and show that they are inequivalent to the |GHZ⟩, |W⟩, or |l, n⟩ (the symmetric Dicke states with l excitations) under SLOCC via the vanishing or not of the polynomials. The absolute values of the polynomials can be considered as entanglement measures.

Two-dimensional dielectric microcavities are of widespread use in microoptics applications. Recently, a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the cavity. In this paper, we extend this trace formula to a dielectric disk with a small scatterer. This system has been introduced for microlaser applications, because it has long-lived resonances with strongly directional far field. We show that its resonance spectrum contains signatures not only of periodic rays but also of diffractive rays that occur in Keller's geometrical theory of diffraction. We compare our results with those for a closed cavity with Dirichlet boundary conditions.

In this paper, we study the scattering theory of the classical hyperbolic Sutherland model associated with the C_n root system. We prove that for any values of the coupling constants the scattering map has a factorized form. As a byproduct of our analysis, we propose a Lax matrix for the rational C_n Ruijsenaars–Schneider–van Diejen model with two independent coupling constants, thereby setting the stage to establish the duality between the hyperbolic C_n Sutherland and the rational C_n Ruijsenaars–Schneider–van Diejen models.

A general class of -algebras can be constructed from the affine sl(N) algebra by (quantum) Drinfeld–Sokolov reduction and are classified by partitions of N. Surface operators in an SU(N) 4D gauge theory are also classified by partitions of N. We argue that instanton partition functions of gauge theories in the presence of a surface operator can also be computed from the corresponding -algebra. We test this proposal by analysing the Polyakov–Bershadsky algebra obtaining results that are in agreement with the known partition functions for SU(3) gauge theories with a so-called simple surface operator. As a byproduct, our proposal implies relations between and algebras.

The Okubo–Weiss field, frequently used for partitioning incompressible two-dimensional (2D) fluids into coherent and incoherent regions, corresponds to the Gaussian curvature of the stream function. Therefore, we consider the differential geometric structures of stream functions and calculate the Gaussian curvatures of some basic flows. We find the following. (I) The vorticity corresponds to the mean curvature of the stream function. Thus, the stream-function surface for an irrotational flow and that for a parallel shear flow correspond to the minimal surface and a developable surface, respectively. (II) The relationship between the coherency and the magnitude of the vorticity is interpreted by the curvatures. (III) Using the Gaussian curvature, stability of single and double point vortex streets is analyzed. The results of this analysis are compared with the well-known linear stability analysis. (IV) Conformal mapping in fluid mechanics is the physical expression of the geometric fact that the sign of the Gaussian curvature does not change in conformal mapping. These findings suggest that the curvatures of stream functions are useful for understanding the geometric structure of an incompressible 2D flow.