low firing rates in a compartmental model neuron a nonlinear relationship is obtained between the membrane potential of a compartmental model neuron as a function of input stimulation this is used to show how shunting effects can lead to low output firing rates in associative memory networks

two qubit separabilities as piecewise continuous functions of maximal concurrence the generic real 1 and complex 2 two qubit states are 9 dimensional and 15 dimensional in nature respectively the total volumes of the spaces they occupy with respect to the hilbert schmidt and bures metrics are obtainable as special cases of formulae of yczkowski and sommers we claim that if one could determine certain metric independent three dimensional eigenvalue parameterized separability functions epsfs s 1 4 1 4 then these formulae could be readily modified so as to yield the hilbert schmidt and bures volumes occupied by only the separable two qubit states and hence associated separability probabilities motivated by analogous earlier analyses of diagonal entry parameterized separability functions we further explore the possibility that such three dimensional epsfs might in turn be expressible as univariate functions of some special relevant variable which we hypothesize to be the maximal concurrence 0 c 1 over spectral orbits extensive numerical results that we obtain are rather closely supportive of this hypothesis both the real and complex estimated epsfs exhibit clearly pronounced jumps of magnitude roughly 50 at c 1 2 as well as a number of additional matching discontinuities

moment based evidence for simple rational valued hilbert schmidt generic 2 2 separability probabilities employing the hilbert schmidt measure we explicitly compute and analyze a number of determinantal product bivariate moments k pt n k n 0 1 2 3 with pt denoting the partial transpose for both generic 9 dimensional two rebit and generic 15 dimensional two qubit 1 density matrices the results are then incorporated into a general formula parameterized by k n and with the case 2 presumptively corresponding to generic 27 dimensional quaternionic systems holding the dyson index like parameter fixed the induced univariate moments pt n and pt n are inputted into a legendre polynomial based least squares probability distribution reconstruction algorithm of provost 2005 mathematica j 9 727 yielding specific separability probability estimates since as the number of inputted moments grows estimates based on the variable pt strongly decrease while ones employing pt strongly increase and converge faster the gaps between upper and lower estimates diminish yielding sharper and sharper bounds remarkably for 2 with the use of 2325 moments a separability probability lower bound 0 999 999 987 as large as is found for 1 based on 2415 moments a lower bound results that is 0 999 997 066 times as large as a simpler still fractional value that had previously been conjectured slater 2007 j phys a math theor 40 14279 furthermore for employing 3310 moments the lower bound is 0 999 955 times as large as a rational value previously considered slater 2010 j phys a math theor 43 195302

alpha divergence is unique belonging to both f divergence and bregman divergence classes a divergence measure between two probability distributions or positive arrays positive measures is a useful tool for solving optimization problems in optimization signal processing machine learning and statistical inference the csiszar f divergence is a unique class of divergences having information monotonicity from which the dual alpha geometrical structure with the fisher metric is derived the bregman divergence is another class of divergences that gives a dually flat geometrical structure different from the alpha structure in general csiszar gave an axiomatic characterization of divergences related to inference problems the kullback leibler divergence is proved to belong to both classes and this is the only such one in the space of probability distributions this paper proves that the alpha divergences constitute a unique class belonging to both classes when the space of positive measures or positive arrays is considered they are the canonical divergences derived from the dually flat geometrical structure of the space of positive measures

a concise formula for generalized two qubit hilbert schmidt separability probabilities we report major advances in the research program initiated in moment based evidence for simple rational valued hilbert schmidt generic 2 2 separability probabilities slater and dunkl 2012 j phys a math theor 45 095305 a highly succinct separability probability function p is put forth yielding for generic nine dimensional two rebit systems 15 dimensional two qubit systems and 27 dimensional two quater nionic bit systems this particular form of p was obtained by qing hu hou by applying zeilberger s algorithm creative telescoping to a fully equivalent but considerably more complicated expression containing six 7f6 hypergeometric functions all with argument that hypergeometric form itself had been obtained using systematic high accuracy probability distribution reconstruction computations these employed 7501 determinantal moments of partially transposed 4 4 density matrices parameterized by from these computations exact rational valued separability probabilities were discernible the integral half integral sequences of 32 rational values then served as input to the mathematica findsequencefunction command from which the initially obtained hypergeometric form of p emerged

on the structure of the body of states with positive partial transpose we show that the convex set of separable mixed states of the 2 2s ystem is a body of a constant height this fact is used to prove that the probability of finding a random state to be separable equals twice the probability of finding a random boundary state to be separable provided that the random states are generated uniformly with respect to the hilbert schmidt euclidean measure an analogous property holds for the set of positive partial transpose states for an arbitrary bipartite system

quantum correlations and distinguishability of quantum states a survey of various concepts in quantum information is given with a main emphasis on the distinguishability of quantum states and quantum correlations covered topics include generalized and least square measurements state discrimination quantum relative entropies the bures distance on the set of quantum states the quantum fisher information the quantum chernoff bound bipartite entanglement the quantum discord and geometrical measures of quantum correlations the article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them and for mathematicians looking for an up to date introductory course on these subjects which are mainly developed in the physics literature

the projective geometry of freudenthal s magic square abstract we connect the algebraic geometry and representation theory associated to freudenthal s magic square we give unified geometric descriptions of several classes of orbit closures describing their hyperplane sections and desingularizations and interpreting them in terms of composition algebras in particular we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial

a priori probabilities of separable quantum states zyczkowski horodecki sanpera and lewenstein zhsl recently proposed a natural measure on the n dimensional quantum systems but expressed surprise when it led them to conclude that for n 2 2 disentangled separable systems are more probable 0 632 0 002 in nature than entangled ones we contend however that zhsl s rejected intuition has in fact a sound theoretical basis and that the a priori probability of disentangled 2 2 systems should more properly be viewed as considerably less than 0 5 we arrive at this conclusion in two quite distinct ways the first based on classical and the second quantum considerations both approaches however replace in whole or part the zhsl product measure by ones based on the volume elements of monotone metrics which in the classical case amounts to adopting the jeffreys prior of bayesian theory only the quantum theoretic analysis which yields the smallest probabilities of disentanglement uses the minimum number of parameters possible that is n2 1 as opposed to n2 n 1 although this over parametrization as recently indicated by byrd should be avoidable however despite substantial computation we are not able to obtain precise estimates of these probabilities and the need for additional possibly supercomputer analyses is indicated particularly so for higher dimensional quantum systems such as the 2 3 ones which we also study here

volumes of conditioned bipartite state spaces we analyze the metric properties of conditioned quantum state spaces these spaces are the convex sets of density matrices that when partially traced over m degrees of freedom respectively yield the given n n density matrix for the case n 2 the volume of equipped with the hilbert schmidt measure can be conjectured to be a simple polynomial of the radius of in the bloch ball remarkably for we find numerically that the probability to find a separable state in is independent of except for pure for the same holds for the probability to find a state with a positive partial transpose in these results are proven analytically for the case of the family of 4 4 x states and thoroughly numerically investigated for the general case the important implications of these findings for the clarification of open problems in quantum theory are pointed out and discussed

the monge metric on the sphere and geometry of quantum states topological and geometrical properties of the set of mixed quantum states in the n dimensional hilbert space are analysed assuming that the corresponding classical dynamics takes place on the sphere we use the vector su 2 coherent states and the generalized husimi distributions to define the monge distance between two arbitrary density matrices the monge metric has a simple semiclassical interpretation and induces a non trivial geometry among all pure states the distance from the maximally mixed state proportional to the identity matrix admits the largest value for the coherent states while the delocalized chaotic states are close to this contrasts the geometry induced by the standard trace hilbert schmidt or bures metrics for which the distance from is the same for all pure states we discuss possible physical consequences including unitary time evolution and the process of decoherence we introduce also a simplified monge metric defined in the space of pure quantum states and more suitable for numerical computation

geometry of canonical correlation on the state space of a quantum system a riemannian metric is defined on the state space of a finite quantum system by the canonical correlation or kubo mori bogoliubov scalar product this metric is infinitesimally induced by the nonsymmetric relative entropy functional or the von neumann entropy of density matrices hence its geometry expresses maximal uncertainty it is proven that the metric is monotone under stochastic mappings however an example shows that it is not the only such riemannian metric this fact is remarkable because in the probabilistic case the markovian monotonicity property characterizes the fisher information metric the essential difference appears in the curvatures of a classical state space and a quantum one a conjecture is made that the scalar curvature is monotone with respect to the more mixed statistical partial order of density matrices furthermore an information inequality resembling the cramer rao inequality of classical statistics is established the inequality provides a lower bound for the c

canonical coset parametrization and the bures metric of the three level quantum systems an explicit parametrization for the state space of an n level density matrix is given the parametrization is based on the canonical coset decomposition of unitary matrices we also compute explicitly the bures metric tensor over the state space of two and three level quantum systems

quantum and fisher information from the husimi and related distributions the two principal immediate influences which we seek to interrelate here upon the undertaking of this study are papers of yczkowski and slomczynski j phys a 34 6689 2001 and of petz and sudar j math phys 37 2262 1996 in the former work a metric the monge one specifically over generalized husimi distributions was employed to define a distance between two arbitrary density matrices in the petz sudar work completing a program of chentsov the quantum analog of the classically unique fisher information monotone metric of a probability simplex was extended to define an uncountable infinitude of riemannian also monotone metrics on the set of positive definite density matrices we pose here the questions of what is the specific unique fisher information metric for the classically defined husimi distributions and how does it relate to the infinitude of quantum metrics over the density matrices of petz and sudar we find a highly proximate small relative entropy relationship bet

statistical properties of random density matrices statistical properties of ensembles of random density matrices are investigated we compute traces and von neumann entropies averaged over ensembles of random density matrices distributed according to the bures measure the eigenvalues of the random density matrices are analysed we derive the eigenvalue distribution for the bures ensemble which is shown to be broader then the quarter circle distribution characteristic of the hilbert schmidt ensemble for measures induced by partial tracing over the environment we compute exactly the two point eigenvalue correlation function

probability measure generated by the superfidelity we study the probability measure on the set of density matrices induced by the metric defined by using superfidelity we give the formula for the probability density of eigenvalues we also study some statistical properties of the set of density matrices equipped with the introduced measure and provide a method for generating density matrices according to the introduced measure

convex bodies of states and maps we give a general solution to the question of when the convex hulls of orbits of quantum states on a finite dimensional hilbert space under unitary actions of a compact group have a non empty interior in the surrounding space of all density operators the same approach can be applied to study convex combinations of quantum channels the importance of both problems stems from the fact that usually only sets with non vanishing volumes in the embedding spaces of all states or channels are of practical importance for the group of local transformations on a bipartite system we characterize maximally entangled states by the properties of a convex hull of orbits through them we also compare two partial characteristics of convex bodies in terms of the largest balls and maximum volume ellipsoids contained in them and show that in general they do not coincide separable states mixed unitary channels and k entangled states are also considered as examples of our techniques

a theory of stochastic resonance in climatic change in this paper we study a one dimensional nonlinear stochastic differential equation when small amplitude long period forcing is applied the equation arises in the theory of the climate of the earth we find that the cooperative effect of the stochastic perturbation and periodic forcing lead to an amplification of the peak of the power spectrum due to a mechanism that we call stochastic resonance a heuristic analysis of the resonance condition is presented and our analytical findings are confirmed by numerical calculations

contraction coefficients for noisy quantum channels generalized relative entropy monotone riemannian metrics geodesic distance and trace distance are all known to decrease under the action of quantum channels we give some new bounds on and relationships between the maximal contraction for these quantities

on the griffiths group of the cubic sevenfold for x a smooth complex projective variety one of the important questions in algebraic geometry is the existence and the behaviour of non trivial subvarieties on x i e subvarieties that are not the intersection of x with a hypersurface in general the approach is as follows we let zd x be the free abelian group generated by the irreducible subvarieties of codimension d in x and introduce various equivalence relation on zd x the problem is then to understand the quotients of zd x under these relations the first and most obvious is homological equivalence but there are other interesting relations namely algebraic and rational equivalence the so called classical cases when d 1 divisors or d dimx 0 cycles are well understood but little is known in general in these cases homological and algebraic equivalence coincide and the group of cycles algebraically equivalent to zero modulo rational equivalence is isomorphic to an abelian variety one of the first result for general d is due to griffiths 8 who showed that homological and algebraic equivalence do not coincide if d 1 dimx the group of cycles on x of codimension d homologous to zero modulo algebraic equivalence is called the d th griffiths group of x and is denoted by gd x we will mainly consider the q vector space g6 x gd x q or equivalently the griffiths group modulo torsion clemens 5 then showed that g6 x can have infinite dimension over q clemens example is given by a general quintic hypersurface in p4 and this opened the way to the construction of other examples see 11 21 131 17

wigner yanase information on quantum state space the geometric approach in the search of appropriate riemannian metrics on quantum state space the concept of statistical monotonicity or contraction under coarse graining has been proposed by chentsov the metrics with this property have been classified by petz all the elements of this family of geometries can be seen as quantum analogs of fisher information although there exists a number of general theorems shedding light on this subject many natural questions also stemming from applications are still open in this paper we discuss a particular member of the family the wigner yanase information using a well known approach that mimics the classical pull back approach to fisher information we are able to give explicit formulas for the geodesic distance the geodesic path the sectional and scalar curvatures associated to wigner yanase information moreover we show that this is the only monotone metric for which such an approach is possible

on quantum statistical inference interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially in step with dramatic new developments in experimental techniques for studying small quantum systems furthermore developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory the present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

linear response characteristics of time dependent time fractional fokker planck equation systems the time fractional fokker planck equation approach is an important tool for modeling subdiffusion when the external field is time modulated two types of time dependent time fractional fokker planck equations have been proposed both reduced to the same time dependent time fractional fokker planck equation when the external field is time uncorrelated the first type is strictly deduced as the continuous limit of the continuous time random walk with time modulated boltzmann jumping weight while the second type is derived by ideally assuming that the jump probabilities can be evaluated at the start of the waiting time prior to jumping for the first time we obtain the linear response characteristic for the first type of the time fractional fokker planck equation systems and for a comparison we revisit the corresponding result for the second type of the time fractional fokker planck equation systems and the similarity and difference between them is discussed with an application example the investigation not only helps in understanding the competition between subdiffusion and time dependent modulation but also has significance in accessing the spectral properties of spontaneous fluctuation and the linear dynamic susceptibility of external perturbation in subdiffusive processes

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

metrics of quantum states in this work we study metrics of quantum states which are natural generalizations of the usual trace metric and bures metric some useful properties of the metrics are proved such as the joint convexity and contractivity under quantum operations our result has a potential application in studying the geometry of quantum states as well as the entanglement detection

volume of the quantum mechanical state space the volume of the quantum mechanical state space over n dimensional real complex and quaternionic hilbert spaces with respect to the canonical euclidean measure is computed and explicit formulae are presented for the expected value of the determinant also in the general setting the case when the state space is endowed with a monotone metric or a pull back metric is also considered we give formulae for the volume of the state space with respect to the given riemannian metric we present the volume of the space of qubits with respect to various monotone metrics it turns out that the volume of the space of qubits can also be infinite we characterize those monotone metrics which generate infinite volume

quantum covariance quantum fisher information and the uncertainty relations in this paper the relation between quantum covariances and quantum fisher informations is studied this study is applied to generalize a recently proved uncertainty relation based on quantum fisher information the proof given here considerably simplifies the previously proposed proofs and leads to more general inequalities

bures and hilbert schmidt 2 2 determinantal moments we seek to gain insight into the nature of the determinantal moments pt n k bures of generic nine dimensional two rebit and fifteen dimensional two qubit systems pt denoting partial transpose such information as it has proved to be in the hilbert schmidt counterpart should be useful employing probability distribution reconstruction inverse procedures in obtaining bures 2 2 separability probabilities the regularizing strategy we first adopt is to plot the ratio of numerically generated ginibre ensemble estimates of the bures moments to the corresponding apparently exactly known hilbert schmidt moments slater and dunkl 2012 j phys a math theor 45 095305 then through a combination of symbolic and numerical computations we obtain strong evidence as to the exact values and underlying patterns of certain bures determinantal moments in particular the first moment average of pt where for the two qubit systems is remarkably the analogous value for the two rebit systems is while in the hilbert schmidt case is the ratio of 3n degree polynomials in k it appears to be the ratio of 5n degree polynomials in k in the bures case

hilbert schmidt volume of the set of mixed quantum states we compute the volume of the convex n2 1 dimensional set n of density matrices of size n with respect to the hilbert schmidt measure the hyper area of the boundary of this set is also found and its ratio to the volume provides information about the structure of n similar investigations are also performed for the smaller set of all real density matrices as an intermediate step we analyse volumes of the unitary and orthogonal groups and of the flag manifolds

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

on the comparison of volumes of quantum states this paper aims to study the volume of an arbitrary subset of the set of n n density matrices the volume is a generalization of the hilbert schmidt volume and the volume induced by partial trace we obtain two side estimates for the volume of in terms of its hilbert schmidt volume the analogous estimates between the bures volume and the volume are also established we employ our results to obtain bounds for the volume of the sets of separable quantum states and of states with positive partial transpose hence our asymptotic results provide answers for questions listed in yczkowski et al 1998 phys rev a 58 883 for large n in the sense of volume

unifying approach to the quantification of bipartite correlations by bures distance the notion of distance defined on the set of states of a composite quantum system can be used to quantify total quantum and classical correlations in a unifying way we provide new closed formulae for classical and total correlations of two qubit bell diagonal states by considering the bures distance complementing the known corresponding expressions for entanglement and more general quantum correlations we thus complete the quantitative hierarchy of bures correlations for bell diagonal states we then explicitly calculate bures correlations for two relevant families of states werner states and rank 2 bell diagonal states highlighting the subadditivity which holds for total correlations with respect to the sum of classical and quantum ones when using bures distance finally we analyse a dynamical model of two independent qubits locally exposed to non dissipative decoherence channels where both quantum and classical correlations measured by bures distance exhibit freezing phenomena in analogy with other known quantifiers of correlations

application of the method of moments for calculating the dynamic response of periodically driven nonlinear stochastic systems it is shown that the method of moments allows one to calculate the first and higher harmonic susceptibilities of nonlinear stochastic systems with high accuracy the dependence of the spectral amplification at the first three harmonics on the noise intensity is studied it is shown that stochastic resonance at the third harmonic occurs at two separate values of the noise intensity for not too large a bias also it is demonstrated that even when the bias is so large that the bistable system turns into a monostable one the stochastic resonant enhancement of the system s sensitivity to the external driving field is still observed at some optimal non zero value of the noise intensity

quantum parameter estimation of a generalized pauli channel we present a quantum parameter estimation theory for a generalized pauli channel d d where the parameter is regarded as a coordinate system of the probability simplex d2 1 we show that for each degree n of extension id n d d n d d n the sld fisher information matrix for the output states takes the maximum when the input state is an n tensor product of a maximally entangled state me d d we further prove that for the corresponding quantum cramer rao inequality there is an efficient estimator if and only if the parameter is m affine in d2 1 these results rely on the fact that the family id me of output states can be identified with d2 1 in the sense of quantum information geometry this fact further allows us to investigate submodels of generalized pauli channels in a unified manner

stochastic resonance in a trapping overdamped monostable system the response of a trapping overdamped monostable system to a harmonic perturbation is analyzed in the context of stochastic resonance phenomenon we consider the dynamics of a brownian particle moving in a piecewise linear potential with a white gaussian noise source based on linear response theory and laplace transform technique analytical expressions of signal to noise ratio snr and signal power amplification spa are obtained we find that the snr is a nonmonotonic function of the noise intensity while the spa is monotonic theoretical results are compared with numerical simulations

stochastic resonance in a bistable piecewise potential analytical solution we consider a bistable piecewise potential subject to both a periodic signal and a random force the output signal to noise ratio snr is calculated for a small amplitude periodic signal of arbitrary frequency and noise strength d our analysis not only recovers the well known non monotonic dependence of snr on d at fixed but also predicts the non monotonic dependence of snr on at fixed d the latter phenomenon does not appear in the commonly assumed adiabatic approximation which applies only at low frequencies

fidelity susceptibility and quantum fisher information for density operators with arbitrary ranks abstract taking into account the density matrices with non full ranks we show that the fidelity susceptibility is determined by the support of the density matrix combining with the corresponding expression of the quantum fisher information we rigorously prove that the fidelity susceptibility is proportional to the quantum fisher information as this proof can be naturally extended to the full rank case this proportional relation is generally established for density matrices with arbitrary ranks furthermore we give an analytical expression of the quantum fisher information matrix and show that it can also be represented in the density matrix s support

minimal entropy of states emerging from noisy quantum channels we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel which is modeled by the action of a completely positive trace preserving or stochastic map we provide strong support for the conjecture that this minimal entropy is additive namely that the minimum entropy can be achieved when product states are transmitted explicitly we prove that for a tensor product of two unital stochastic maps on qubit states using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states we give a separate argument based on the geometry of the image of the set of density matrices under stochastic maps which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps we also show that the maximal norm of the output states is multiplicative for most product maps on n qubit states including all those for which at least one map is unital for the class of unital channels on c sup 2 we show that additivity of minimal entropy implies that the holevo see ieee trans inform theory vol 44 p 269 73 1998 and russian math surv p 1295 1331 1999 capacity of the channel is additive over two inputs achievable with orthogonal states and equal to the shannon capacity this implies that superadditivity of the capacity is possible only for nonunital channels

some extremal contractions between smooth varieties arising from projective geometry we construct explicit examples of elementary extremal contractions both birational and of fiber type from smooth projective n dimensional varieties with n geq 4 onto smooth projective varieties arising from classical projective geometry and defined over sufficiently small fields not necessarily algebraically closed the examples considered come from particular special homaloidal and subhomaloidal linear systems which are usually degenerations of general phenomena classically investigated by bordiga severi todd room fano semple and tyrrell and more recently by ein and shepherd barron the first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of mathbf p m with 3 leq m leq 5 we get another series of examples by considering special cubic hypersurfaces through some surfaces in mathbf p 5 or some 3 folds in mathbf p 7 having one apparent double point the last examples come from an intriguing birational elementary extremal contraction in dimension 6 studied by semple and tyrrell and fully described in the last section of the paper

trace distance from the viewpoint of quantum operation techniques in the present paper the trace distance is exposed within the quantum operations formalism the definition of the trace distance in terms of a maximum over all quantum operations is given it is shown that for any pair of different states there are an uncountably infinite number of maximizing quantum operations conversely for any operation of the described type there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation the behaviour of the trace distance under considered operations is studied relations and distinctions between the trace distance and the sine distance are discussed

on the bures volume of separable quantum states we obtain two sided estimates for the bures volume of an arbitrary subset of the set of n n density matrices in terms of the hilbert schmidt volume of that subset for general subsets our results are essentially optimal for large n as applications we derive in particular nontrivial lower and upper bounds for the bures volume of sets of separable states and for sets of states with positive partial transpose

ratios of maximal concurrence parameterized separability functions and generalized peres horodecki conditions the probability that a generic real complex or quaternionic two qubit state is separable can be considered to be the sum of three contributions one is from those states that are absolutely separable that is those which can not be entangled by unitary transformations for which the maximal concurrence over spectral orbits cmax is zero the other two contributions are from the states for which and for which we have previously arxiv 0805 0267 found exact formulas for the absolutely separable contributions in terms of the hilbert schmidt metric over the quantum states and here advance hypotheses as to the exact contributions for a crucial element in understanding the two contributions for cmax 0 is the nature of the ratio r of the cmax parameterized separability function for the complex states to the square of the comparable function for the real states both such functions having clearly displayed jump discontinuities at for the ratio r appears to be of the form 1 kcmax except near while for there is strong numerical evidence that it equals 2 thus according to the dyson index pattern of random matrix theory related phenomena also occur for the minimally degenerate two qubit states and the qubit qutrit states our results have immediate application to the computation of separability probabilities in terms of other metrics such as the bures minimal monotone metric the paper begins with continuous embeddings of the separability probability question in terms of four metrics of interest using generalized peres horodecki conditions

stochastic resonance at higher harmonics in monostable systems the nonlinear response of a system subject to a periodic force in the presence of noise is investigated it is shown that a physical system of a general type manifests stochastic resonance at higher harmonics the stochastic resonance curves are calculated for several monostable systems higher harmonic stochastic resonance is measured on a monostable part of a domain wall pinned by a magnetic defect in a ferrite garnet film

random bures mixed states and the distribution of their purity ensembles of random density matrices determined by various probability measures are analysed a simple and efficient algorithm to generate at random density matrices distributed according to the bures measure is proposed this procedure may serve as an initial step in performing the bayesian approach to quantum state estimation based on the bures prior we study the distribution of purity of random mixed states the moments of the distribution of purity are determined for quantum states generated with respect to the bures measure this calculation serves as an exemplary application of the deform and study approach in the theory of integrable systems leading to one of painleves transcendents

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

k3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds the main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the fano variety of lines of a cubic 4 fold f in general different from f z f a general k3 surface s of genus 8 determines uniquely a pair of cubic 4folds the apolar cubic f s and the dual pfaffian cubic f s or for simplicity f and f as beauville and donagi have shown the fano variety tf of lines on the cubic f is isomorphic to the hilbert scheme hilb2 s of length two subschemes of s the first main result of this paper is that hilb2 s parametrizes the variety vsp f 10 of presentations of the cubic form f with f z f as a sum of 10 cubes which yields an isomorphism between nff and vsp f 10 furthermore we show that vsp f 10 sets up a 6 10 correspondence between f and ff the main result follows by a deformation argument 1 pfaffian and apolar cubic 4 folds associated to k3 surfaces of genus 8 1 1 let v be a 6 dimensional vector space over c fix a basis e e5 for v then ei a ej for 0 i j aijei aej e a2v a skewsymmetric matrix m g aij with aji aj with pliicker coordinates xij the embedding of the grassmannian g g 2 v in p14 p a2v is then precisely the locus of rank 2 skewsymmetric 6 x 6 matrices 0 x01 x02 x03 x04 x05 x0i 0 x12 x13 x14 x15 m x02 x12 0 x23 x24 x25 x03 x13 x23 0 x34 x35 x04 x14 x24 x34 0 x45 x05 x15 x25 x35 x45 0 since the sum of two rank 2 matrices has rank at most 4 and any rank 4 skewsymmetric matrix is the sum of two rank 2 skewsymmetric matrices the secant variety of g is the cubic hypersurface k defined by the 6 x 6 pfaffian m of the matrix m the dual variety of g in p14 p a2v is a cubic hypersurface k k cf 10 k is the secant variety of g g v 2 the grassmannian of rank 2 quotient spaces of v and of course g c g received by the editors july 5 1999 2000 mathematics subject classification primary 14j70 secondary 14m15 14n99 2000 american mathematical society

geometry of quantum systems density states and entanglement various problems concerning the geometry of the space of hermitian operators on a hilbert space are addressed in particular we study the canonical poisson and riemann jordan tensors and the corresponding foliations into k hler submanifolds it is also shown that the space of density states on an n dimensional hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state thus the space of rank k states k 1 n is a smooth manifold of real dimension 2nk k2 1 and this stratification is maximal in the sense that every smooth curve in viewed as a subset of the dual to the lie algebra of the unitary group at every point must be tangent to the strata it crosses for a quantum composite system i e for a hilbert space decomposition an abstract criterion of entanglement is proved

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

stochastic resonance in piecewise potentials different characteristics of an output signal average position population average energy are calculated for a particle moving in a piecewise potential and subject to external periodic and random forces particular emphasis has been placed on the dependence of these characteristics on the strength of the noise and the frequency of an external field an external periodic force strives to equalize the populations of the discrete levels or even to reverse the populations for the space extended systems the populations of the potential wells in a one barrier system subject to oscillation of the wells can either decrease or increase compared with the field free case depending on the frequencies of the external field all these changes of populations induced by an external field have some resemblance to the similar quantum mechanical problems

uncertainty principle and quantum fisher information ii heisenberg and schrodinger uncertainty principles give lower bounds for the product of variances var a var b if the observables a b are not compatible namely if the commutator a b is not zero in this paper we prove an uncertainty principle in schrodinger form where the bound for the product of variances var a var b depends on the area spanned by the commutators i a and i b with respect to an arbitrary quantum version of the fisher information

the ppt square conjecture holds generically for some classes of independent states let be a random pure state on where is a random unit vector uniformly distributed on the sphere in let be a random induced state whose distribution is and let be a random induced state following the same distribution independent of let be a random state induced by the entanglement swapping of and we show that the empirical spectrum of converges almost surely to the marcenko pastur law with parameter c 2 as and as an application we prove that the state is separable generically if are ppt entangled

subnormalized states and trace nonincreasing maps we investigate the set of completely positive trace nonincreasing linear maps acting on the set mn of mixed quantum states of size n extremal point of this set of maps are characterized and its volume with respect to the hilbert schmidt hs euclidean measure is computed explicitly for an arbitrary n the spectra of partially reduced rescaled dynamical matrices associated with trace nonincreasing completely positive maps belong to the n cube inscribed in the set of subnormalized states of size n as a by product we derive the measure in mn induced by partial trace of mixed quantum states distributed uniformly with respect to the hs measure in mn2

generating random density matrices we study various methods to generate ensembles of random density matrices of a fixed size n obtained by partial trace of pure states on composite systems structured ensembles of random pure states invariant with respect to local unitary transformations are introduced to analyze statistical properties of quantum entanglement in bi partite systems we analyze the distribution of schmidt coefficients of random pure states such a distribution is derived in the case of a superposition of k random maximally entangled states for another ensemble obtained by performing selective measurements in a maximally entangled basis on a multi partite system we show that this distribution is given by the fuss catalan law and find the average entanglement entropy a more general class of structured ensembles proposed containing also the case of bures forms an extension of the standard ensemble of structureless random pure states described asymptotically as n by the marchenko pastur distribution

dyson indices and hilbert schmidt separability functions and probabilities a confluence of numerical and theoretical results leads us to conjecture that the hilbert schmidt separability probabilities of the 15 and 9 dimensional convex sets of complex and real two

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

generalized euler angle parametrization for su n in a previous paper 1 an euler angle parametrization for su 4 was given here we present the derivation of a generalized euler angle parametrization for su n the formula for the calculation of the haar measure for su n as well as its relation to marinov s volume formula for su n 2 3 will also be derived as an example of this parametrization s usefulness the density matrix parametrization and invariant volume element for a qubit qutrit three qubit and two three state systems also known as two qutrit systems 4 will also be given

bures volume of the set of mixed quantum states we compute the volume of the n2 1 dimensional set n of density matrices of size n with respect to the bures measure and show that it is equal to that of an n2 1 dimensional hyper hemisphere of radius 1 2 for n 2 we obtain the volume of the uhlmann hemisphere s3 4 we find also the area of the boundary of the set n and obtain analogous results for the smaller set of all real density matrices an explicit formula for the bures hall normalization constants is derived for an arbitrary n

on cubic hypersurfaces of dimension seven and eight cubic sevenfolds are examples of fano manifolds of calabi yau type we study them in relation with the cartan cubic the e 6 invariant cubic in pp 26 we show that a generic cubic sevenfold x can be described as a linear section of the cartan cubic in finitely many ways to each such cartan representation we associate a rank nine vector bundle on x with very special cohomological properties in particular it allows to define auto equivalences of the non commutative calabi yau threefold associated to x by kuznetsov finally we show that the generic eight dimensional section of the cartan cubic is rational

geometries of quantum states the quantum analog of the fisher information metric of a probability simplex is searched and several riemannian metrics on the set of positive definite density matrices are studied some of them appeared in the literature in connection with cramer rao type inequalities or the generalization of the berry phase to mixed states they are shown to be stochastically monotone here all stochastically monotone riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them a class of metrics can be extended to pure states and a constant multiple of the fubini study metric appears in the extension

explicit formulae for the bures metric the aim of this paper is to derive explicit formulae for the computation of the riemannian bures metric g on the manifold of finite dimensional nonsingular density matrices this riemannian metric introduced by uhlmann generalizes the fubini study metric to mixed states and is the infinitesimal version of the bures distance several formulae are known for computing the bures metric in low dimensions the formulae presented in this paper allow for computing in finite dimensions without any diagonalization procedures the first equations we give are essentially of the form where is a matrix of invariants of a further formula is adapted to the local orthogonal decomposition at generic points

observing fluctuating spectral density of subdiffusive overdamped brownian particles in periodic potentials the aim of this paper is to develop a simple and efficient method for observing the fluctuating spectral density of subdiffusive brownian motion in an overdamped periodic potential for exploring the subdiffusive property in frequency domain based on the general frame of linear response theory for subdiffusive fractional fokker planck equation systems an explicit relation between fluctuating spectral density and linear dynamical susceptibility is deduced and then a method of moments based on the expansion of trigonometric functions is proposed for calculating the linear dynamic susceptibility with the linear dynamic susceptibility available the fluctuating spectral density is finally obtained the numerical results demonstrate that subdiffusion weakens coherent oscillations in the periodic system but enhances aperiodic components our observation embodies the fact of the mittag leffler residence time distribution with an infinite mean in the subdiffusive process from the frequency domain

connections and metrics respecting purification of quantum states standard purification interlaces hermitian and riemannian metrics on the space of density operators with metrics and connections on the purifying hilbert schmidt space we discuss connections and metrics which are well adopted to purification and present a selected set of relations between them a connection as well as a metric on state space can be obtained from a metric on the purification space we include a condition with which this correspondence becomes one to one our methods are borrowed from elementary representation and fiber space theory we lift as an example solutions of a von neumann equation write down holonomy invariants for cyclic ones and add noise to a curve of pure states

on the geometry of mixed states and the fisher information tensor in this paper we will review the co adjoint orbit formulation of finite dimensional quantum mechanics and in this framework we will interpret the notion of quantum fisher information index and metric following previous work of part of the authors who introduced the definition of fisher information tensor we will show how its antisymmetric part is the pullback of the natural kostant kirillov souriau symplectic form along some natural diffeomorphism in order to do this we will need to understand the symmetric logarithmic derivative as a proper 1 form settling the issues about its very definition and explicit computation moreover the fibration of co adjoint orbits seen as spaces of mixed states is also discussed

an explicit computation of the bures metric over the space of n dimensional density matrices the aim of this paper is to provide a method for explicit computation of the bures metric over the space of n level quantum system based on the coset parametrization of density matrices

calculation of the unitary part of the bures measure for n level quantum systems we use the canonical coset parameterization and provide a formula with the unitary part of the bures measure for non degenerate systems in terms of the product of even euclidean balls this formula is shown to be consistent with the sampling of random states through the generation of random unitary matrices

two quantum analogues of fisher information from a large deviation viewpoint of quantum estimation we discuss two quantum analogues of the fisher information the symmetric logarithmic derivative fisher information and kubo mori bogoljubov fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of consistent superefficient estimators as another comparison it is shown that the difference between them is characterized by the change of the order of limits

exact variance of von neumann entanglement entropy over the bures hall measure the bures hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing in this work we study the statistical behavior of quantum entanglement over the bures hall ensemble as measured by von neumann entropy the average von neumann entropy over such an ensemble has been recently obtained whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average the starting point of the calculations is the connection between correlation functions of the bures hall ensemble and those of the cauchy laguerre ensemble the derived variance formula together with the known mean formula leads to a simple but accurate gaussian approximation of the distribution of von neumann entropy of finite size systems this gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems

stochastic resonance in monostable systems the first observations of noise induced enhancements and phase shifts of a weak periodic signal characteristics signatures of stochastic resonance sr are reported for a monostable system the results are shown to be in good agreement with a theoretical description based on linear response theory and the fluctuation dissipation theorem it is argued that sr is a general phenomenon that can in principle occur for any underdamped nonlinear oscillator

on skew information in this paper we show that skew information introduced by wigner and yanase which is a natural informational extension of variance for pure states can be interpreted as a measure of quantum uncertainty by virtue of skew information we establish a new uncertainty relation in the spirit of schrodinger which incorporates both incompatibility encoded in the commutator and correlations encoded in a new correlation measure related to skew information between observables and moreover is stronger than the conventional ones

completely positive mappings and mean matrices abstract some functions f r r induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices moreover such a function f can define a linear mapping j d f 1 m n m n on matrices which is basic in the constructions of monotone metrics the present subject is to check the complete positivity of j d f 1 in the case of a few concrete functions f this problem has been motivated by applications in quantum information

on the monotonicity of scalar curvature in classical and quantum information geometry we study the monotonicity under mixing of the scalar curvature for the geometries on the simplex of probability vectors from the results obtained and from numerical data we are led to some conjectures about quantum geometries and wigner yanase dyson information finally we show that this last conjecture implies the truth of the petz conjecture about the monotonicity of the scalar curvature of the bogoliubov kubo mori monotone metric