a statistical mechanical description of quantum entanglement we present a description of finite dimensional quantum entanglement based on a study of the space of all convex decompositions of a given density matrix on this space we construct a system of real polynomial equations describing separable states we further study this system using methods of statistical mechanics as an example we finally apply our techniques to werner states of two qubits and obtain a sufficient criterion for separability

the inverse eigenvalue problem for nonnegative matrices abstract let a be a nonnegative matrix with spectrum 1 2 m and b be a nonnegative matrix with spectrum 1 2 n where 1 is the perron eigenvalue of b furthermore let a maximal diagonal element of a be greater than or equal to 1 in the article we construct a nonnegative matrix c with spectrum 1 2 m 2 n this construction enables us to obtain several results on how to determine new realizable lists from known realizable lists

nonuniversal critical behaviour in planar ising models with extended radially symmetric defects using conformal mapping and finite size scaling the authors determine the local magnetic exponent near a defect characterized by couplings of the form j r j 1 a r in accordance with phenomenological considerations they find that it depends continuously on the amplitude a both for a defect in the interior and on the boundary

asymptotic stability and stabilizability of special classes of discrete time positive switched systems abstract in this paper we consider discrete time positive switched systems switching among autonomous subsystems characterized either by monomial matrices or by circulant matrices necessary and sufficient conditions are provided guaranteeing either global uniform asymptotic stability or stabilizability i e the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence such conditions lead to simple algorithms that allow to easily detect under suitable conditions whether a given positive switched system is not stabilizable

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

a map of sufficient conditions for the real nonnegative inverse eigenvalue problem abstract the real nonnegative inverse eigenvalue problem rniep is the problem of determining necessary and sufficient conditions for a list of real numbers to be the spectrum of an entrywise nonnegative matrix a number of sufficient conditions for the existence of such a matrix are known in this paper in order to construct a map of sufficient conditions we compare these conditions and establish inclusion relations or independency relations between them

a unified view on compensation criteria in the real nonnegative inverse eigenvalue problem a connection is established between the problem of characterizing all possible real spectra of entrywise nonnegative matrices the so called real nonnegative inverse eigenvalue problem and a combinatorial process consisting in repeated application of three basic manipulations on sets of real numbers given realizable sets i e sets which are spectra of some nonnegative matrix each of these three elementary transformations constructs a new realizable set this defines a special kind of realizability called c realizability and this is closely related to the idea of compensation after observing that the set of all c realizable sets is a strict subset of the set of realizable ones we show that it strictly includes in particular all sets satisfying several previously known sufficient realizability conditions in the literature furthermore the proofs of these conditions become much simpler when approached from this new point of view

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

elgenvalues of nonnegative matrices abstract in a partial continuation of work by fiedler some spectral properties of symmetric nonnegative matrices are extended to general nonnegative matrices

existence and construction of nonnegative matrices with prescribed spectrum abstract we consider the following inverse spectrum problem for nonnegative matrices given a set of real numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum in particular by the use of a relevant theorem of brauer we obtain new simple sufficient conditions for the problem to have a solution moreover we can always construct a solution matrix which is nonnegative generalized stochastic

realizability criterion for the symmetric nonnegative inverse eigenvalue problem abstract let 1 2 n a set of real numbers the real nonnegative inverse eigenvalue problem rniep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix if there exists a nonnegative matrix a with spectrum we say that is realized by a many realizability criteria for the existence of such a matrix a are known this paper shows that a realizability criterion given by the author which contains both kellogg s realizability criterion and borobia s realizability criterion is sufficient for the existence of an n n symmetric nonnegative matrix with prescribed spectrum

block matrices and guo s index for block circulant matrices with circulant blocks abstract in this paper we deal with circulant and partitioned into n by n circulant blocks matrices and introduce spectral results concerning this class of matrices the problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated along the paper we call realizable list if its elements are the eigenvalues of a nonnegative matrix the guo s index 0 of a realizable list is the minimum spectral radius such that the list up to the initial spectral radius together with 0 is realizable the guo s index of block circulant matrices with circulant blocks is obtained and in consequence necessary and sufficient conditions concerning the niep nonnegative inverse eigenvalue problem for the realizability of some spectra are given

semi infinite two dimensional ising model with marginally inhomogeneous couplings and conformal invariance the authors consider a semi infinite two dimensional ising model with nearest neighbour couplings that deviate from the bulk critical couplings by al 2 where l is the distance from the surface the surface critical exponents of this system are non universal under a conformal mapping onto a strip of width l the al 1 inhomogeneity transforms into a l pi sin pi l l 1 for the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit the analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance

local critical behaviour at aperiodic surface extended perturbation in the ising quantum chain the surface critical behaviour of the semi infinite one dimensional quantum ising model in a transverse field is studied in the presence of an aperiodic surface extended modulation the perturbed couplings are distributed according to a generalized fredholm sequence leading to a marginal perturbation and varying surface exponents the surface magnetic exponents are calculated exactly whereas the expression of the surface energy density exponent is conjectured from a finite size scaling study the system displays surface order at the bulk critical point above a critical value of the modulation amplitude it may be considered as a discrete realization of the hilhorst van leeuwen model

existence and construction of nonnegative matrices with complex spectrum abstract the following inverse spectrum problem for nonnegative matrices is considered given a set of complex numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum our work is motivated by a relevant theoretical result of guo wuwen linear algebra appl 266 1997 261 theorem 2 1 there exists a real parameter 0 max 2 j n j such that the problem has a solution if and only if 1 0 in particular we discuss how to compute 0 and the solution matrix a for certain class of matrices a sufficient condition for the problem to have a solution is also derived

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

realizable lists on a class of nonnegative matrices abstract a square matrix of order n with n 2 is called permutative matrix when all its rows are permutations of the first row in this paper recalling spectral results for partitioned into 2 by 2 symmetric blocks matrices sufficient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given in particular we study niep and pniep when some complex elements in the lists under consideration have non zero imaginary part realizability regions for nonnegative permutative matrices are obtained a guo s realizability preserving perturbations result is obtained

realizable lists via the spectra of block matrices abstract in this paper we present spectral results for matrices partitioned into higher order blocks where not all blocks are necessarily square using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative matrix are obtained and the corresponding matrix is constructed in particular spectral results for symmetric matrices are derived a result on guo perturbations for permutative matrices on given lists is generalized

realizable lists via the spectra of structured matrices abstract a square matrix of order n with n 2 is called a permutative matrix or permutative when all its rows up to the first one are permutations of precisely its first row in this paper the spectra of a class of permutative matrices are studied in particular spectral results for matrices partitioned into 2 by 2 symmetric blocks are presented and using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative permutative matrix are obtained and the corresponding permutative matrices are constructed guo perturbations on given lists are exhibited

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

perturbing non real eigenvalues of non negative real matrices abstract let b i c b i c 4 n be the spectrum of an entry non negative matrix and t 0 laffey t j laffey perturbing non real eigenvalues of nonnegative real matrices electron j linear algebra 12 2005 73 76 has shown that 2 t b t i c b t i c 4 n is also the spectrum of some nonnegative matrix laffey 2005 has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t in this paper a rank two perturbation is used to deduce an explicit and constructive proof for all t 0

on the nonnegative eigenvalue problem abstract 1 1 n is the spectrum of a stochastic matrix of order n 1 if and only if there exists some real matrix b of order n with spectrum 1 n and some n simplex s r n containing the origin such that bs s this result appears implicitly in ciarlet s work and it provides us with a geometrical tool to obtain sufficient conditions for the nonnegative eigenvalue problem we employ it here to generalize the sufficient conditions given by kellogg

the nonnegative inverse eigenvalue problem abstract let 1 n be a list of complex numbers the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix our purpose is to give an overview of the niep its history modern results and subproblems with particular emphasis on symmetric realizing matrices a substantial bibliography is included

spectral decomposition of real circulant matrices abstract this paper presents spectral decompositions i e eigendecompositions and singular value decompositions of four types of real circulant matrices right and left circulants whose elements topple from right to left or from left to right respectively as well as skew right and skew left circulants whose elements change their sign when toppling are analyzed the inherent periodicity of circulant matrices means that they are closely related to fourier analysis and group theory this relationship is utilized in the spectral decompositions of this paper

radial fredholm perturbation in the two dimensional ising model and gap exponent relation we consider concentric circular defects in the two dimensional ising model which are distributed according to a generalized fredholm sequence i e at exponentially increasing radii this type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation the critical exponent of the local magnetization is obtained through finite size scaling using a corner transfer matrix approach in the extreme anisotropic limit it varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial hilhorst van leeuwen model through a conformal mapping of the system onto a strip the gap exponent relation is shown to remain valid for such an aperiodic defect

some results on symmetric circulant matrices and on symmetric centrosymmetric matrices we derive a readily computable sufficient condition for the existence of a nonnegative symmetric circulant matrix having a prescribed spectrum moreover we prove that any set 1 2 n is the spectrum of real symmetric centrosymmetric matrices s1 and s2 such that for s1 an eigenvector corresponding to 1 is the all ones vector and for s2 an eigenvector corresponding to 1 is the vector with components 1 and 1 alternately the proof is constructive then we derive an improved condition on k k 1n such that s1 1e where e is a stochastic symmetric centrosymmetric matrix finally we propose an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices all the numerical procedures are based on the use of the fast fourier transform

perturbations on constructible lists preserving realizability in the niep and questions of monov abstract in 5 the authors showed that if 1 2 2 4 n is realizable where 1 is the perron eigenvalue and 2 is non real then so too is 1 4 t 2 t 2 t 4 n they asked if 4 can be replaced by 1 or 2 or what is the least possible multiple c 1 of t in order for this perturbation to be realizable in 2 the authors showed that for n 4 one can find certain spectra for which the result holds when c 1 provided t is small in this paper we show that c 2 is best possible and we construct a realizing matrix for c 2 when t is sufficiently small we also address some questions of monov concerning the realizability of the derivative of a realizable polynomial and if such a polynomial must have positive power sums

moments of non gaussian wigner distributions and a generalized uncertainty principle i the single mode case the non negativity of the density operator of a state is faithfully coded in its wigner distribution and this coding places on the moments of the wigner distribution constraints arising from the non negativity of the density operator working in a monomial basis for the algebra of operators on the hilbert space of a bosonic mode we formulate these constraints in a canonically covariant form which is both concise and explicit since the conventional uncertainty relation is such a constraint on the first and second moments our result constitutes a generalization of the same to all orders the structure constants of in the monomial basis are shown to be essentially the su 2 clebsch gordan coefficients our results have applications in quantum state reconstruction using optical homodyne tomography and when generalized to the n mode case which will be done in the second part of this work will have applications also for continuous variable quantum information systems involving non gaussian states

hilbert s 17th problem and the quantumness of states a state of a quantum system can be regarded as classical quantum with respect to measurements of a set of canonical observables if and only if there exists does not exist a well defined positive phase space distribution the so called glauber sudarshan p representation we derive a family of classicality criteria that requires that the averages of positive functions calculated using p representation must be positive for polynomial functions these criteria are related to hilbert s 17th problem and have physical meaning of generalized squeezing conditions alternatively they may be interpreted as nonclassicality witnesses we show that every generic nonclassical state can be detected by a polynomial that is a sum of squares of other polynomials we introduce a very natural hierarchy of states regarding their degree of quantumness which we relate to the minimal degree of a sum of squares polynomial that detects them

construction of nonnegative symmetric matrices with given spectrum abstract let 1 n be the spectrum of a nonnegative symmetric matrix a with the perron eigenvalue 1 a diagonal entry c and let 1 m be the spectrum of a nonnegative symmetric matrix b with the perron eigenvalue 1 we show how to construct a nonnegative symmetric matrix c with the spectrum 1 max 0 1 c 2 n 2 m

applications of a brauer theorem in the nonnegative inverse eigenvalue problem abstract a result by brauer which shows how to modify one single eigenvalue of a matrix without changing any of the remaining eigenvalues plays a relevant role in the study of the nonnegative inverse eigenvalue problem niep perfect in a long time ignored paper 1955 presents an extension of this result which shows how to modify r eigenvalues of a matrix of order n r n via a rank r perturbation without changing any of the n r remaining eigenvalues by using this extension perfect gives a realizability criterion for the real niep which is not contained in soto s realizability criterion in this work by extending perfect s result we give a new realizability criterion for the real niep which contains soto s criterion thus this new realizability criterion appears to be the most general sufficient condition for the real niep so far we also contribute to the solution of the symmetric niep for n 5

guo perturbation for symmetric nonnegative circulant matrices abstract guo w guo eigenvalues of nonnegative matrices linear algebra appl 266 1997 261 270 sets the question if the list 1 2 n is symmetrically realizable that is is the spectrum of a symmetric nonnegative matrix and t 0 whether or not the list t 1 t 2 t 3 n is also symmetrically realizable in this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant we also give a necessary and sufficient condition for to be the spectrum of a nonnegative left circulant matrix

an inverse eigenvalue problem for nonnegative matrices abstract this paper gives a solution to an inverse eigenvalue problem of nonnegative matrices which was proposed by miroslav fiedler

an inverse eigenvalue problem for symmetric and normal matrices abstract some new sufficient conditions are found for n real numbers to be the spectrum of some n n symmetric nonnegative matrix and for n complex numbers to be the spectrum of some n n normal nonnegative matrix