local asymptotic normality for finite dimensional quantum systems previous results on local asymptotic normality lan for qubits 16 19 are extended to quantum systems of arbitrary finite dimension d lan means that the quantum statistical model consisting of n identically prepared d dimensional systems with joint state rho otimes n converges as n to a statistical model consisting of classical and quantum gaussian variables with fixed and known covariance matrix and unknown means related to the parameters of the density matrix remarkably the limit model splits into a product of a classical gaussian with mean equal to the diagonal parameters and independent harmonic oscillators prepared in thermal equilibrium states displaced by an amount proportional to the off diagonal elements as in the qubits case 16 lan is the main ingredient in devising a general two step adaptive procedure for the optimal estimation of completely unknown d dimensional quantum states this measurement strategy shall be described in a forthcoming paper 18

asymptotic performance of optimal state estimation in qubit system we derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus it is also proven that this bound can be achieved in any statistical model in the qubit system moreover we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models that is in such a statistical model the quantum correlation can improve the accuracy of the estimation in an asymptotic setting

quantum estimation for non differentiable models state estimation is a classical problem in quantum information in optimization of an estimation scheme to find a lower bound to the error of the estimator is a very important step so far all the proposed tractable lower bounds use a derivative of the density matrix however sometimes we are interested in quantities with singularity e g concurrence etc in this paper lower bounds to a mean square error of an estimator are derived for a quantum estimation problem without smoothness assumptions our main idea is to replace the derivative by difference as is done in classical estimation theory we applied the inequalities to several examples and derived an optimal estimator for some of them

tracking pattern evolution through extended center manifold reduction and singular perturbations in this paper we develop an extended center manifold reduction method a methodology to analyze the formation and bifurcations of small amplitude patterns in certain classes of multi component singularly perturbed systems of partial differential equations we specifically consider systems with a spatially homogeneous state whose stability spectrum partitions into eigenvalue groups with distinct asymptotic properties one group of successive eigenvalues in the bifurcating group are widely interspaced while the eigenvalues in the other are stable and cluster asymptotically close to the origin along the stable semi axis the classical center manifold reduction provides a rigorous framework to analyze destabilizations of the trivial state as long as there is a spectral gap of sufficient width when the bifurcating eigenvalue becomes commensurate to the stable eigenvalues clustering close to the origin the center manifold reduction breaks down moreover it cannot capture subsequent bifurcations of the bifurcating pattern through our methodology we formally derive expressions for low dimensional manifolds exponentially attracting the full flow for parameter combinations that go beyond those allowed for the classical center manifold reduction i e to cases in which the spectral gap condition no longer can be satisfied our method also provides an explicit description of the flow on these manifolds and thus provides an analytical tool to study subsequent bifurcations our analysis centers around primary bifurcations of transcritical type that can be either of co dimension 1 or 2 in two and three component pde systems we employ our method to study bifurcation scenarios of small amplitude patterns and the possible appearance of low dimensional spatio temporal chaos we also exemplify our analysis by a number of characteristic reaction diffusion systems with disparate diffusivities

centre manifold reduction and the stuart landau equation for fluid motions the centre manifold reduction to derive the stuart landau equation is examined a double expansion in terms of the fourier series and linear eigenfunctions is introduced in hydrodynamic equations a centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the stuart landau equation through a formal expansion in linear eigenfunctions the latter equation is shown to be equivalent with the one derived by the method of multiple scales numerical coefficients involved in the quintic stuart landau equation are evaluated for plane poiseuille flow convection in a vertical slot and rayleigh benard convection in all the cases the coefficients converge as a dimension of the ode system increases and approach the numerical values obtained by the method of multiple scales

a new approach to the cramer rao type bound of the pure state model this paper sheds light on non commutativity in quantum theory as regards theoretical estimation in it we calculate the quantum cramer rao type bound for many cases by use of a newly proposed powerful technique we also discuss the use of collective measurement in statistical estimation

asymptotic estimation theory for a finite dimensional pure state model the optimization of measurement for n samples of pure states are studied the error of the optimal measurement for n samples is asymptotically compared with the one of the maximum likelihood estimators from n data given by the optimal measurement for one sample

methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions the equivalence between the method of centre manifold and the method of multiple scales was examined when they are applied to the weakly nonlinear stability analysis for fluid motions the relationship between the landau equations of the seventh order which were reduced from the navier stokes equations by applying these two methods is clarified how we should define the disturbance amplitude to obtain the complete equivalence between them is discussed

stochastic stability of coupled oscillators simple and combination resonances in coupled oscillators under a random parametric excitation are studied an efficient numerical method for obtaining the regions of mean square stability is proposed based on the method stability diagrams for different values of the excitation parameter and for four form of the coupling parameters are presented it is found that the random excitation can have a stabilizing effect under certain conditions

attaining the ultimate precision limit in quantum state estimation we derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non degenerate our results hold under an assumption called local asymptotic covariance which is weaker than unbiasedness or local unbiasedness the derivation is based on an analysis of the limiting distribution of the estimator s deviation from the true value of the parameter and takes advantage of quantum local asymptotic normality a useful asymptotic characterization of identically prepared states in terms of gaussian states we first prove our results for the mean square error of a special class of models called d invariant and then extend the results to arbitrary models generic cost functions and global state estimation where the unknown parameter is not restricted to a local neighbourhood of the true value the extension includes a treatment of nuisance parameters i e parameters that are not of interest to the experimenter but nevertheless affect the precision of the estimation as an illustration of the general approach we provide the optimal estimation strategies for the joint measurement of two qubit observables for the estimation of qubit states in the presence of amplitude damping noise and for noisy multiphase estimation

optimal estimation of qubit states with continuous time measurements we propose an adaptive two step strategy for the estimation of mixed qubit states we show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit local minimax optimality means that given n identical qubits there exists no estimator which can perform better than the proposed estimator on a neighborhood of size n 1 2 of an arbitrary state in particular it is asymptotically bayesian optimal for a large class of prior distributions we present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits the crucial ingredient of the result is the concept of local asymptotic normality or lan for qubits this means that for large n the statistical model described by n identically prepared qubits is locally equivalent to a model with only a classical gaussian distribution and a gaussian state of a quantum harmonic oscillator the term local refers to a shrinking neighborhood around a fixed state 0 an essential result is that the neighborhood radius can be chosen arbitrarily close to n 1 4 this allows us to use a two step procedure by which we first localize the state within a smaller neighborhood of radius n 1 2 and then use lan to perform optimal estimation

the real and complex lorenz equations in rotating fluids and lasers abstract the lorenz equations are derived systematically from amplitude equations of weakly nonlinear dispersively unstable physical systems near criticality when weak dissipation is added this derivation is only valid if the undamped neutral curve is not destabilised by the addition of weak dissipation the addition of extra weak dispersive effects make some of the coefficients complex and yields a complex set of lorenz equations both sets of equations are derived in examples in laser optics and baroclinic instability

moment stability for linear systems with a random parametric excitation moment stability for linear systems with a nonwhite parametric noise is considered a method of reduction of the study of this stability to the study of stability for large scale matrices is proposed mean square stability diagrams for random harmonic oscillator are presented

comparison of some approaches to stability of a sdof system under random parametric excitation we give a rigorous comparison of the asymptotic stochastic averaging method and the moment lyapunov exponent approach in assessing the mean square stability through an example of a single degree of freedom system under small periodic parametric excitation with random phase modulation the comparison is based on an efficient numerical procedure to determine the regions of mean square stability

the 1 2 spatial resonance on a hexagonal lattice in two layered rayleigh benard problems this paper investigates pattern formation resulting from the interaction between steady modes with wavenumbers in the ratio 1 2 a two layered rayleigh benard problem with infinite extent is examined two layers are separated by a non deformable and thin dividing plate this physical set up is known to provide an exact 1 2 resonance between critical modes restricting the wavevectors of interacting modes on a hexagonal lattice we derive 12 dimensional amplitude equations up to cubic order among steady solutions of the equations the stable ones are hexagons and mixed hexagons travelling waves corresponding to relative equilibria are also found to be stable depending on parameter values

multiparameter estimation in networked quantum sensors we introduce a general model for a network of quantum sensors and we use this model to consider the following question when can entanglement between the sensors and or global measurements enhance the precision with which the network can measure a set of unknown parameters we rigorously answer this question by presenting precise theorems proving that for a broad class of problems there is at most a very limited intrinsic advantage to using entangled states or global measurements moreover for many estimation problems separable states and local measurements are optimal and can achieve the ultimate quantum limit on the estimation uncertainty this immediately implies that there are broad conditions under which simultaneous estimation of multiple parameters cannot outperform individual independent estimations our results apply to any situation in which spatially localized sensors are unitarily encoded with independent parameters such as when estimating multiple linear or nonlinear optical phase shifts in quantum imaging or when mapping out the spatial profile of an unknown magnetic field we conclude by showing that entangling the sensors can enhance the estimation precision when the parameters of interest are global properties of the entire network

probing a subcritical instability with an amplitude expansion an exploration of how far one can get abstract we explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite wavelength instability we do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers this is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic laminar state we apply the method to two model equations a subcritical generalization of the swift hohenberg equation and a novel extension of the kuramoto sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate from infinity and that can nevertheless be probed perturbatively we explore the reliability and robustness of such an expansion with a particular focus on the use of these methods for determining the existence and approximate properties of finite amplitude stationary solutions such methods obviously are to be used with caution the expansions are often only asymptotic approximations and if they converge their radius of convergence may be small nevertheless expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results

on the nonlinear dynamics of a saline boundary layer formed by throughflow near the surface of a porous medium we consider gravitational instability of saline boundary layers observed at the subsurface of salt lakes this boundary layer is the result of the convective transport induced by the evaporation at the horizontal surface of a confined porous medium when this upward transport is balanced by salt dispersion a steady state boundary layer is formed however this boundary layer can be unstable when perturbed this results in complex groundwater motion and density fields the aim of this paper is to investigate the existence of finite amplitude solutions describing these resulting patterns both the number of solutions and their structure their stability and their dependency on the system rayleigh and peclet numbers for this purpose we construct a low dimensional dynamical system a reduced model by projecting the nonlinear model equations onto a relatively small set of eigenfunctions of the problem linearized at criticality the galerkin projection approach is complicated by the fact that the problem under consideration is non self adjoint due to the existing evaporation this implies that the eigenfunctions do not form an orthogonal set and therefore the adjoint eigenfunctions are used for the projection the reduced model is constructed in such a way that it is capable of providing solutions in the strongly nonlinear regime as well convergence of these solutions towards the fully nonlinear model results is shown by means of direct numerical simulations further the reduced model seems to partly capture the complex nonlinear behavior as seen in hele shaw experiments by wooding et al r a wooding s w tyler i white p a anderson convection in groundwater below an evaporating salt lake 2 evolution of fingers or plumes water resour res 33 6 1997 1219 1228 the physical transition mechanism that explains the occurrence of some observed bifurcation types is presented as well

comparison between the cramer rao and the mini max approaches in quantum channel estimation in a unified viewpoint in quantum channel estimation we compare the cramer rao and the mini max approaches which gives the bayesian bound in the group covariant model for this purpose we introduce the local asymptotic mini max bound whose maximum is shown to be equal to the asymptotic limit of the mini max bound it is shown that the local asymptotic mini max bound is strictly larger than the cramer rao bound in the phase estimation case while both bounds coincide when the minimum mean square error decreases with the order o frac 1 n we also derive a sufficient condition so that the minimum mean square error decreases with the order o frac 1 n

fourier analytic approach to quantum estimation of group action this article proposes a unified method to estimation of group action by using the inverse fourier transform of the input state the method provides optimal estimation for commutative and non commutative groups with and without energy constraint the proposed method can be applied to projective representations of non compact groups as well as of compact groups this paper addresses the optimal estimation of mathbb r u 1 su 2 so 3 and mathbb r 2 with heisenberg representation under a suitable energy constraint

the application of centre manifolds to amplitude expansions ii infinite dimensional problems abstract the theory of centre manifolds for infinite dimensional systems is described with emphasis on the practical computational aspects of applying the theory to near critical problems and in particular to computation of the centre manifold the calculations are illustrated by detailed analyses of two specific problems

explicit formula for the holevo bound for two parameter qubit state estimation problem the main contribution of this paper is to derive an explicit expression for the fundamental precision bound the holevo bound for estimating any two parameter family of qubit mixed states in terms of quantum versions of fisher information the obtained formula depends solely on the symmetric logarithmic derivative sld the right logarithmic derivative rld fisher information and a given weight matrix this result immediately provides necessary and sufficient conditions for the following two important classes of quantum statistical models the holevo bound coincides with the sld cramer rao bound and it does with the rld cramer rao bound one of the important results of this paper is that a general model other than these two special cases exhibits an unexpected property the structure of the holevo bound changes smoothly when the weight matrix varies in particular it always coincides with the rld cramer rao bound for a certain choice of the weight matrix several examples illustrate these findings

evaluating the holevo cramer rao bound for multiparameter quantum metrology only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed this is due to the incompatibility of observables the fundamental bound for multiparameter quantum estimation is the holevo cram er rao bound hcrb whose evaluation has so far remained elusive for finite dimensional systems we recast its evaluation as a semidefinite program with reduced size for rank deficient states we show that it also satisfies strong duality we use this result to study phase and loss estimation in optical interferometry and three dimensional magnetometry with noisy multiqubit systems for the former we show that in some regimes it is possible to attain the hcrb with the optimal single copy measurement for phase estimation for the latter we show a nontrivial interplay between the hcrb and incompatibility and provide numerical evidence that projective single copy measurements attain the hcrb in the noiseless 2 qubit case

degenerate 1 2 steady state mode interaction mhd flow in a vertical slot abstract degenerate bifurcations resulting from 1 2 mode interaction between quasi neutral steady modes in mhd natural convection have been investigated on a weakly nonlinear basis in the limit of small prandtl number and small magnetic prandtl number two types of nonlinearly degenerate situations are found to take place for relatively low hartmann numbers 4 0432 and 8 7701 the amplitude equations of the quintic order are derived by means of the center manifold reduction the entire picture of the bifurcation is obtained for the actual numerical values of the coefficients in the amplitude equations a fully nonlinear numerical analysis of the hydrodynamic equations is also carried out the local characteristics predicted by the amplitude equations are found to be justified

hexagons and triangles in the rayleigh benard problem quintic order equations on a hexagonal lattice on a weakly nonlinear basis we revisit the pattern formation problem in the boussinesq convection for which nonlinear terms of the quadratic order are known to vanish from amplitude equations it