reduced dynamics for entangled initial state of the full density operator starting from the schrodinger equation for a system and a reservoir we construct a quantum dynamical map that describes reduced dynamics we work with an arbitrary initial state for the system and reservoir and thus refrain from performing any factorizations the map is found to be nonlinear not completely positive and not unique to investigate the physical implications the bloch equations are constructed if the freedom the new map offers us is fully exploited then the classic inequality 2 for the damping coefficients must be replaced by a weaker condition

martingale type convergence of modular automorphism groups on von neumann algebras abstract the strong convergence of modular automorphism groups and connes cocycle derivatives is discussed under an increasing or decreasing net of von neumann subalgebras and faithful semifinite normal weights on a von neumann algebra

dynamical entropy of space translations of car and ccr algebras with respect to quasi free states we compute the dynamical entropy in the sense of connes narnhofer and thirring of space translations of the car and ccr algebras in dimensional continuous spaces with respect to invariant quasi free states it turns out that the dynamical entropies are equal to the corresponding mean entropies of the systems under consideration computing the mean entropies explicitly we derive the entropy formulas for the systems

quantum markov states on cayley trees abstract it is known that any locally faithful quantum markov state qms on one dimensional setting can be considered as a gibbs state associated with hamiltonian with commuting nearest neighbor interactions in our previous results we have investigated quantum markov states qms associated with ising type models with competing interactions which are expected to be qms but up to now there is no any characterization of qms over trees we notice that these qms do not have one dimensional analogues hence results of related to one dimensional qms are not applicable therefore the main aim of the present paper is to describe of qms over cayley trees namely we prove that any qms associated with localized conditional expectations can be realized as integral of product states w t r a gibbs measure moreover it is established that any locally faithful qms associated with localized conditional expectations can be considered as a gibbs state corresponding to hamiltonians on the cayley tree with commuting competing interactions

une nouvelle definition de l entropie dynamique des systemes non commutatifs an alternative definition of entropy for non commutative systems equivalent to the one by a connes h narnhofer and w thirring is given it is based on the concepts of conditional entropy and stationary couplings with classical systems it allows to prove that any quantum dynamical system with singular spectrum has zero entropy

quantum fluctuation relations for the lindblad master equation an open quantum system interacting with its environment can be modeled under suitable assumptions as a markov process described by a lindblad master equation in this work we derive a general set of fluctuation relations for systems governed by a lindblad equation these identities provide quantum versions of jarzynski hatano sasa and crooks relations in the linear response regime these fluctuation relations yield a fluctuation dissipation theorem fdt valid for a stationary state arbitrarily far from equilibrium for a closed system this fdt reduces to the celebrated callen welton kubo formula

markov property and strong additivity of von neumann entropy for graded quantum systems the quantum markov property is equivalent to the strong additivity of von neumann entropy for graded quantum systems the additivity of von neumann entropy for bipartite graded systems implies the statistical independence of states however the structure of markov states for graded systems is different from that for tensor product systems which have trivial grading for three composed graded systems we have u 1 gauge invariant markov states whose restriction to the marginal pair of subsystems is nonseparable

generalized quantum onsager coefficients from a symmetrized campbell hausdorff expansion the analogue of the classical onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite dimensional hilbert space the dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of gorini kossakowski sudarshan type the basic spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics to this end an appropriate lie series is constructed from a particular symmetrization procedure applied to the ordinary campbell hausdorff expansion in this way only hermitian contributions by higher order commutators are generated which allow an identification with so called generalized onsager coefficients the explicit derivations concentrate on second third and fourth order coefficients whereas complete detailed expressions are worked out for second and third order in a suitable coherence vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the lie algebra of su n as an illustration an application to generalized bloch equations for two level systems is studied where the onsager like expansion can be compared with exact results for entropy production we find that convergence is good even for rather large deviations between initial and final state if the calculation includes second and third order coefficients only the formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes

on the existence of quantum subdynamics it is shown using only elementary operator algebra that an open quantum system coupled to its environment will have a subdynamics reduced dynamics as an exact consequence of the reversible dynamics of the composite system only when the states of system and environment are uncorrelated furthermore it is proved that for a finite temperature the kms condition for the lowest order correlation function cannot be reproduced by any type of linear subdynamics except the reversible hamiltonian one of a closed system the first statement can be seen as a particular case of a more general theorem of takesaki on the properties of conditional expectations in von neumann algebras the concept of subdynamics used here allows for memory effects no assumption is made of a markov property for dynamical systems based on commutative algebras of observables the subdynamics always exists as a stochastic process in the random variable defining the open subsystem

retarded and instantaneous evolution equations of macroobservables in non equilibrium statistical mechanics abstract the relations between evolution equations ee for macroobservables of the retarded and instantaneous time convolutionless type are discussed from a general point of view far from equilibrium processes are included in using linear operator representations of nonlinear ee conditions under which a given retarded ee can be memory renormalized are formulated and the renormalized versions of nonequilibrium theories of robertson and grabert are derived memory kernels containing long time tails which cannot be renormalized are considered explicitly the corresponding instantaneous ee exhibit the existence of two different regimes of decay the transition between them taking place at a critical time t k ln where characterizes the strength of the long time tail contribution if using a logarithmic time scale at very large times the relaxation is described by a markovian equation with a universal transport coefficient which is independent of the microscopic properties of the system

poisson boundary of the dual of suq n we prove that for any non trivial product type action of su q n 0 q 1 on an itpfi factor n the relative commutant is isomorphic to the algebra l of bounded measurable functions on the quantum flag manifold this is equivalent to the computation of the poisson boundary of the dual discrete quantum group the proof relies on a connection between the poisson integral and the berezin transform our main technical result says that a sequence of berezin transforms defined by a random walk on the dominant weights of su n converges to the identity on the quantum flag manifold

investigation of continuous time quantum walk via modules of bose mesner and terwilliger algebras the continuous time quantum walk on the underlying graphs of association schemes has been studied via the algebraic combinatorics structures of association schemes namely semi simple modules of their bose mesner and terwilliger algebras it is shown that the terwilliger algebra stratifies the graph into a d 1 disjoint union of strata which is different from the stratification based on distance except for distance regular graphs in underlying graphs of association schemes the probability amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes such that the amplitudes of observing the continuous time quantum walk on all sites belonging to a given stratum are the same therefore there are at most d 1 different observing probabilities the importance of association scheme in continuous time quantum walk is shown by some worked out examples such as arbitrary finite group association schemes followed by symmetric sn dihedral d2m and cyclic groups at the end it is shown that the highest irreducible representations of terwilliger algebras pave the way to use the spectral distributions method of jafarizadeh and salimi 2005 preprint quant ph 0510174 in studying quantum walk on some rather important graphs called distance regular graphs

classes of conditional expectations over von neumann algebras let n m be von neumann algebras and e m n an conditional expectation mapping for a state of n an extension gye of with respect to e is described the relation e e defined to hold if gye gye for every is an equivalence relation the family of equivalence classes possesses an affine structure and shows analogy with the normal state space of a von neumann algebra

quantum theory of dissipative processes the markov approximation revisited adopting the standard mathematical framework for describing reduced dynamics we derive two formal identities for the density operator of an open quantum system each of these is equivalent to the old nakajima zwanzig equation the first identity is local in time it contains the inverse of the dynamical map which govern the evolution of the density operator we indicate a time interval on which this inverse exists the second identity constitutes a suitable starting point for going beyond the markov approximation in a controlled way on the basis of the bloch equations we argue once more that in studying quantum dissipation one has to pay attention to the von neumann conditions in the nakajima zwanzig equation we make the first born approximation the ensuing master equation possesses the correct weak coupling limit while proving this rather obvious but at the same time important statement we elucidate the mathematical methods which underlie the weak coupling limit moving to a two dimensional hilbert space we find that both for short and for long times our approximate master equation respects the von neumann conditions assuming exponential decay for correlation functions we propose a physical limit in which the solutions for the density operator become markovian in character we confirm the well known statement that as seen from a macroscopic standpoint the system starts from an effective initial condition the approach to equilibrium is exponential the accessory relaxation constants can differ from the usual bloch parameters and by more than 50

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

dilation of a dissipative quantum dynamical system to a quantum markov process on considere une c etoile algebre 7b a qui est une algebre de von neumann 7b m et p t est un semigroupe d applications completement positives normales preservant l identite avec un etat normal invariant on obtient un processus de markov non commutatif au sens de accardi avec une dilatation du semigroupe vers un groupe d automorphisme implante unitaire

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

conditional expectations in von neumann algebras and a theorem of takesaki conditional expectations play an important role in classical probability theory in the general context of von neumann algebras they were impliciteiy used by von neumann 41 chap ii and by dixmier 14 nakamura and turumaru 27 and umegaki 36 391 introduced an axiomatic definition of the concept of conditional expectation in the framework of von neumann or c algebras and established many properties of these objects especially in the context of von neumann algebras with a finite trace their starting point was the characterization given by moy 26 of the classical conditional expectations as operators on spaces of measurable functions tomiyama showed 33 that conditional expectations in the sense of the above mentioned authors can be characterized as norm one projection in c algebras the importance of norm one projection in the classification problem of von neumann algebras was recognized by hakeda and tomiyama 23 and subsequent research on this argument confirmed the usefulness of these objects this line of thought culminated in the fundamental work of connes 121 in which approximately finite von 245 0022 1236 82 020245 29 02 00 o

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

separation of fast and slow variables for a linear system by the method of multiple time scales abstract a matrix method introduced by geigenmuller titulaer and felderhof for separating fast and slow motions of a system of first order linear differential equations is shown to arise in a natural way from a multiple time scales expansion the general method is then used to derive the smoluchowski equation and its initial data for brownian motion in a strong magnetic field from the corresponding kramers chandrasekhar initial value problem

dynamical entropy of c algebras and von neumann algebras the definition of the dynamical entropy is extended for automorphism groups ofc algebras as an example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density

dilations of semigroups on von neumann algebras and noncommutative lp spaces abstract we prove that any weak continuous semigroup t t t 0 of factorizable markov maps acting on a von neumann algebra m equipped with a normal faithful state can be dilated by a group of markov automorphisms analogous to the case of a single factorizable markov operator which is an optimal result we also give a version of this result for strongly continuous semigroups of operators acting on noncommutative l p spaces and examples of semigroups to which the results of this paper can be applied our results imply the boundedness of the mcintosh s h functional calculus of the generators of these semigroups on the associated noncommutative l p spaces generalising some previous work from junge le merdy and xu finally we also give concrete dilations for poisson semigroups which are even new in the case of r n

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

resistance distances in composite graphs the resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor let g h g h g h g h g h gh and g h be the join product composition direct product strong product corona and rooted product of two graphs g and h respectively in this paper formulae for resistance distances of these composite graphs are given in terms of parameters of the parent graphs and some properties are established explicit formulae are obtained for resistance distances of some classes of graphs including rook graphs diagonal mesh graphs and generalized double graphs

factorization and dilation problems for completely positive maps on von neumann algebras we study factorization and dilation properties of markov maps between von neumann algebras equipped with normal faithful states i e completely positive unital maps which preserve the given states and also intertwine their automorphism groups the starting point for our investigation has been the question of existence of non factorizable markov maps as formulated by c anantharaman delaroche we provide simple examples of non factorizable markov maps on m n mathbb c for all n 3 as well as an example of a one parameter semigroup t t t 0 of markov maps on m 4 mathbb c such that t t fails to be factorizable for all small values of t 0 as applications we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum birkhoff conjecture as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little grothendieck inequality

calculating effective resistances on underlying networks of association schemes recently in the work of jafarizadeh et al j phys a math theor 40 4949 2007 e print arxiv 0705 2480 calculation of effective resistances on distance regular networks was investigated where in the first paper the calculation was based on stratification and stieltjes functions associated with the network whereas in the latter one a recursive formula for effective resistances was given based on the christoffel darboux identity in this paper evaluation of effective resistances on more general networks that are underlying networks of association schemes is considered where by using the algebraic combinatoric structures of association schemes such as stratification and bose mesner algebras an explicit formula for effective resistances on these networks is given in terms of the parameters of the corresponding association schemes moreover we show that for particular underlying networks of association schemes with diameter d such that the adjacency matrix a possesses d 1 distinct eigenvalues

evolution matrix in a coherence vector formulation for quantum markovian master equations of n level systems quantum markovian master equation in n dimensions are systematically transformed into vector form by using the lie algebra of the special unitary group su n density operators are represented through coherence vectors and the infinitesimal kossakowski generator of completely positive quantum dynamical semigroups appears as a real evolution matrix that is not completely diagonalisable in general a complete classification of the spectrum of the latter and its jordan canonical form together with all types of associated stationary states is given in detail the results of this analysis are particularly suited for computations in practical applications

entanglement entropy in the spinless free fermion model and its application to the graph isomorphism problem in this paper the entanglement entropy is investigated in the ground state of a spinless free fermion hamiltonian where its hopping matrix is given by the adjacency matrix of the graph the bipartite entanglement entropy is calculated from the eigenvalues of the correlation matrix the entanglement entropy and mutual information are calculated for some scalable graphs including a complete graph and three sets of strongly regular graphs srgs in addition the volume law scaling has been shown for these graphs then we use the entanglement entropy as a tool for studying the graph isomorphism problem in some non isomorphic pairs of graphs in order to distinguish them from each other two graphs are isomorphic when they are related to each other by relabeling the graph vertices we study the graph isomorphism in non isomorphic srgs distance regular graphs and pseudo distance regular graphs most pairs of non isomorphic srgs can be distinguished from each other by using fermionic entanglement entropy

translation invariant pure state on b j zmd j c and its split property in this paper we prove that a real lattice symmetric reflection positive translation invariant pure state of b j zmd j c admits split property if and only if its two point spatial correlation functions decay exponentially we use amalgamated representation of cuntz algebras to represent two point spatial correlation functions on an augmented hilbert space the underling symmetries and reflection positive property of the pure state make it possible to investigate its split and decaying two point correlation functions properties as spectral properties of a contractive self adjoint operator on the augmented hilbert space haag duality property of the pure state is crucially used in the analysis

investigation of continuous time quantum walk on root lattice an and honeycomb lattice abstract the continuous time quantum walk ctqw on root lattice a n known as hexagonal lattice for n 2 and honeycomb one is investigated by using spectral distribution method to this aim some association schemes are constructed from abelian group z m n and two copies of finite hexagonal lattices such that their underlying graphs tend to root lattice a n and honeycomb one as the size of the underlying graphs grows to infinity the ctqw on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on terwilliger algebra where we get the required results for root lattice a n and honeycomb one from large enough underlying graphs moreover by using the stationary phase method the long time behavior of ctqw on infinite hexagonal lattice is approximated with finite ones for finite distances from the origin while for large distances the scaling behavior of the probability distribution is deduced also it is shown that the probability amplitudes at finite times possess the point group symmetry in the sense that two lattice points have the same probability amplitudes if and only if they belong to the same orbit stratum of the graph of the point group of the lattice apart from physical results it is shown that the bose mesner algebras of our constructed association schemes called n variable p polynomial can be generated by n commuting generators where raising flat and lowering operators as elements of terwilliger algebra are associated with each generator a system of n variable orthogonal polynomials which are special cases of generalized gegenbauer polynomials is constructed where the probability amplitudes are given by integrals over these polynomials or their linear combinations finally the supersymmetric structure of finite honeycomb lattices is revealed

convergence of multitime correlation functions in the weak and singular coupling limits for a system coupled to a thermal bath we prove the convergence of the multitime correlation functions of system observables in the weak and singular coupling limits the limiting correlation functions are given by the quantum regression law therefore our result implies that in the limit the dynamics of the system are governed by a quantum stochastic process in the sense of lindblad

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

non homogeneous quantum markov states and quantum markov fields abstract the program relative to the investigation of quantum markov states for general one dimensional spin models is carried on following a strategy developed in the last years in such a way the emerging structure is fully clarified this analysis is a starting point for the solution of the basic still open problem concerning the construction of a satisfactory theory of quantum markov fields i e quantum markov processes with multi dimensional indices

regularization of quantum relative entropy in finite dimensions and application to entropy production the fundamental concept of relative entropy is extended to a functional that is regular valued also on arbitrary pairs of nonfaithful states of open quantum systems this regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution on this basis a generalized formula for entropy production is proposed the applicability of which is tested in models of irreversible processes the dynamics of the latter is determined by either markovian or non markovian master equations and involves all types of states

the modular symmetry of markov maps abstract a state preserving automorphism of a von neumann algebra induces a canonical unitary operator on the gns hilbert space of the state which fixes the vacuum this unitary commutes with both the modular operator of the state and its modular conjugation we prove an extension of this result for state preserving unital completely positive maps

non markovian behavior in low temperature damping an application of the averaging method abstract we give a description of the reduced dynamics of an open quantum system whose initial state is in general correlated with the surroundings the averaging method whose mathematical theory was developed in another paper is used to construct a series expansion of the reduced dynamics in powers of the ratio 1 0 of the characteristic relaxation times 1 and 0 of the surroundings and of the system respectively to first order in 1 0 we obtain the well known markovian approximation which corresponds to the weak coupling limit starting from second order in 1 0 there appear various kinds of non markovian effects including initial slips which have attracted much attention in the recent literature the relevance of these higher order corrections stems from the fact that the characteristic time 1 cannot be much smaller than h kt for quantum surroundings at temperature t

on ergodic theorems for free group actions on noncommutative spaces we extend in a noncommutative setting the individual ergodic theorem of nevo and stein concerning measure preserving actions of free groups and averages on spheres s 2 n of even radius here we study state preserving actions of free groups on a von neumann algebra a and the behaviour of s 2 n x for x in noncommutative spaces l p a for the cesaro means this problem was solved by walker our approach is based on ideas of bufetov we prove a noncommutative version of rota alternierende verfahren theorem to this end we introduce specific dilations of the powers of some noncommutative markov operators

strong martingale convergence of generalized conditional expectations on von neumann algebras accardi and cecchini generalized the concept of conditional expectations on von neumann algebras in this paper we give some conditions for strong convergence of increasing or decreasing martingales of accardi and cecchini s conditional expectations introduction the study of conditional expectations and martingale convergence theorems in the operator algebraic framework was initiated by umegaki s earlier works 14 15 since then the martingale convergence theorems on von neumann algebras have been developed by several authors for example the almost sure type martingale convergence was obtained by lance 9 and dang ngoc 4 the strong martingale convergence was completed in 13 by one of the authors the conditional expectation of a von neumann algebra m onto a von neumann subalgebra n does not generally exist given a faithful normal state or semifinite weight p on m indeed its existence is equivalent to the invariance of n under the modular automorphism group associated with p see 12 to recover this drawback of noncommutative conditional expectations accardi and cecchini 1 generalized the concept of conditional expectations on von neumann algebras by using the tomita takesaki theory 11 accardi and cecchini s conditional expectation e m 3 n with respect to p always exists but is not necessarily a projection onto n and lacks a useful property that c axb ac x b for a b e n and x e m araki 2 established the convergence of modular operators and modular conjugations given an increasing net of von neumann subalgebras and a faithful normal state by extending the arguments in 2 we have obtained in 8 the martingale type convergence of modular automorphism groups under an increasing or decreasing net of von neumann subalgebras and a faithful normal semifinite weight on the same lines we investigate in this paper the strong convergence of martingales of accardi and cecchini s conditional expectations called here generalized conditional expectations in 1 of this paper we state the definition of the generalized conditional expectation with respect to a faithful normal semifinite weight and give easy technical received by the editors march 15 1983 and in revised form may 22 1983 1980 mathematics subject classification primary 46l10

quantum chaos an entropy approach a new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics it generalizes other definitions of entropy in both the classical and quantum cases the kolmogorov sinai ks entropy is obtained for a classical system and the sharp measurement instrument for a quantum system and a coherent states instrument a new quantity coherent states entropy is defined it may be used to measure chaos in quantum mechanics the following correspondence principle is proved the upper limit of the coherent states entropy of a quantum map as 0 is less than or equal to the ks entropy of the corresponding classical map chaos umpire sits and by decision more imbroils the fray by which he reigns next him high arbiter chance governs all john milton paradise lost book ii

separation of fast and slow variables for a linear system with time dependent coefficients abstract a systematic perturbation theory introduced by geigenmuller titulaer and felderhof for separating fast and slow motions of a set of first order linear differential equations is generalized to the case when the coefficients in the equation system vary slowly in time the conditions under which this system has a reduced causal description are given the general technique is used to derive the smoluchowski equation and its initial data from the initial value problem for the kramers chandrasekhar equation in the case when the external force is time dependent

pure inductive limit state and kolmogorov s property abstract let b t be a c dynamical system where t t t be a semigroup of injective endomorphism and be an t invariant state on the c subalgebra b and t is either non negative integers or real numbers the central aim of this exposition is to find a useful criteria for the inductive limit state b t b canonically associated with to be pure we achieve this by exploring the minimal weak forward and backward markov processes associated with the markov semigroup on the corner von neumann algebra of the support projection of the state to prove that kolmogorov s property a mohari markov shift in non commutative probability j funct anal 199 2003 189 209 of the markov semigroup is a sufficient condition for the inductive state to be pure as an application of this criteria we find a sufficient condition for a translation invariant factor state on a one dimensional quantum spin chain to be pure this criteria in a sense complements criteria obtained in o bratteli p e t jorgensen a kishimoto r f werner pure states on o d j operator theory 43 1 2000 97 143 a mohari markov shift in non commutative probability j funct anal 199 2003 189 209 as we could go beyond lattice symmetric states

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

numerical continuation methods for studying periodic travelling wave wavetrain solutions of partial differential equations abstract periodic travelling waves wavetrains are an important solution type for many partial differential equations in this paper i review the use of numerical continuation for studying these solutions i discuss the calculation of the form and stability of a given periodic travelling wave and the calculation of boundaries in a two dimensional parameter plane for wave existence and stability i also describe the automated implementation of these numerical continuation procedures via the software package wavetrain http www ma hw ac uk wavetrain i conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves

martingale convergence of generalized conditional expectations abstract we prove that the martingale convergence theorem for generalized conditional expectations in von neumann algebras holds in the weak topology without restrictions the situation is therefore different fom the strong topology case where there are restrictive conditions which distinguish between increasing and decreasing sequences of von neumann algebras moreover known couterexamples show that in the decreasing case the strong martingale convergence theorem might not hold

limit theorem for continuous time quantum walk on the line concerning a discrete time quantum walk x t d with a symmetric distribution on the line whose evolution is described by the hadamard transformation it was proved by the author that the following weak limit theorem holds x t d t ensuremath rightarrow dx ensuremath pi 1 ensuremath x 2 sqrt 1 ensuremath 2 x 2 as t ensuremath rightarrow ensuremath infty the present paper shows that a similar type of weak limit theorem is satisfied for a continuous time quantum walk x t c on the line as follows x t c t ensuremath rightarrow dx ensuremath pi sqrt 1 ensuremath x 2 as t ensuremath rightarrow ensuremath infty these results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete and continuous time classical random walks y t sqrt t ensuremath rightarrow e ensuremath x 2 2 dx sqrt 2 ensuremath pi as t ensuremath rightarrow ensuremath infty the work deals also with the issue of the relationship between discrete and continuous time quantum walks this topic subject of a long debate in the previous literature is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases

determinism and randomness in quantum dynamics a class of models is considered where a finite or infinite quantum dynamical system in a stationary state is probed by sequences of observations acting on a specified finite subsystem the whole set of such experiments is described by a time ordered causal and stationary quantum correlation kernel it is shown that any such kernel can be decomposed in a unique way into a convex combination of two kernels here called the regular and singular components a singular system has a strong deterministic property the predictability of the future from the knowledge of the past is limited only by the inevitable indeterminism of quantum measurements furthermore in this case the full set of correlation functions of arbitrary time order and hence the dynamical system itself is determined by the causal kernel finite systems and infinite systems satisfying the kms condition at finite temperature are of this type in the regular case the dynamics contains a shift there is a genuine asymptotic randomness and the dynamical system cannot be reconstructed in a unique way from the causal kernel non trivial quantum markov processes are shown to belong to this class

the martin boundary of a discrete quantum group we consider the markov operator p on a discrete quantum group given by convolution with a q tracial state in the study of harmonic elements x p x x we define the martin boundary a it is a separable c algebra carrying canonical actions of the quantum group and its dual we establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on a the c algebraa has a natural time evolution and the unit can always be represented by a kms state any such state gives rise to a u c p map from the von neumann closure of a in its gns representation to the von neumann algebra of bounded harmonic elements which is an analogue of the poisson integral under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual next we apply these results to identify the martin boundary of the dual of suq 2 with the quantum homogeneous sphere of podle this result extends and unifies previous results by ph biane and m izumi

markov dilations on w algebras abstract in this paper a systematic study of markov dilations is begun for completely positive operators on w algebras which leave a faithful normal state invariant it is shown that a minimal markov dilation preserves important properties of the underlying completely positive operator afterwards some results are proved concerning the construction of dilations which lead to markov dilations for large classes of operators finally some of the ideas developed here are applied to the study of a simple example over the 2 2 matrices

reduced dynamics need not be completely positive the reduced dynamics of a quantum system in contact with a reservoir is generally thought to be completely positive this is certainly true if product initial conditions are used to define the dynamics we show that with correlated initial conditions it need not be so in this case the dynamics can properly be defined only on a subset of initial system states extension by linearity to all possible initial states is trivially possible but the extension may not be physically realizable and may not even be positive let alone completely positive

wave train selection behind invasion fronts in reaction diffusion predator prey models abstract wave trains or periodic travelling waves can evolve behind invasion fronts in oscillatory reaction diffusion models for predator prey systems although there is a one parameter family of possible wave train solutions in a particular predator invasion a single member of this family is selected sherratt 1998 13 has predicted this wave train selection using a system that is a valid approximation near a supercritical hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal away from a hopf bifurcation or if the diffusion coefficients differ somewhat these predictions lose accuracy we develop a more general wave train selection prediction for a two component reaction diffusion predator prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front this prediction retains accuracy farther away from a hopf bifurcation and can also be applied when the predator and prey diffusion coefficients are unequal we illustrate the selection prediction with its application to three models of predator invasions

theory of spin relaxation processes abstract the relaxation properties of a spin system weakly coupled to lattice degrees of freedom are described using an equation of motion for the spin density matrix this equation is derived using a general weak coupling theory which has been previously developed to second order in the weak coupling parameter the results are in agreement with those obtained by bloch wangsness and redfield but the derivation does not make use of second order perturbation theory for short times the derivation can be extended beyond second order and ensures that the spin density matrix relaxes to its exact equilibrium form to the appropriate order in the weak coupling parameter

phase transitions for quantum markov chains associated with ising type models on a cayley tree the main aim of the present paper is to prove the existence of a phase transition in quantum markov chain qmc scheme for the ising type models on a cayley tree note that this kind of models do not have one dimensional analogous i e the considered model persists only on trees in this paper we provide a more general construction of forward qmc in that construction a qmc is defined as a weak limit of finite volume states with boundary conditions i e qmc depends on the boundary conditions our main result states the existence of a phase transition for the ising model with competing interactions on a cayley tree of order two by the phase transition we mean the existence of two distinct qmc which are not quasi equivalent and their supports do not overlap we also study some algebraic property of the disordered phase of the model which is a new phenomena even in a classical setting

invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave abstract periodic wave trains are the generic solution from for oscillatory reaction diffusion equations in one space dimension it has been shown previously that invasive wavefronts generate behind them a wave train with a different speed from that of the invasion sherratt physica d 70 1994 370 382 in this paper the mechanism of wave train generation is studied in detail for systems of two reaction diffusion equations close to a supercritical hopf bifurcation in the kinetics with equal diffusion coefficients a combination of analytical and numerical evidence is presented suggesting that the invasive front and wave train are separated by a modulated travelling wave of phase gradient in which phase singularities occur periodically this calculation leads to a prediction of the amplitude and speed of the wave train generated by invasion

systematic elimination of fast variables in linear systems we consider linear systems in which some of the variables evolve on a much faster time scale than the remaining ones by means of a systematic perturbation theory in the time scale ratio we extract a reduced dynamics in terms of the slow variables only valid after an initial transient period the procedure provides a systematic extension of the usual adiabatic elimination scheme and gives corrections to it we also find an expression for the long time behavior of the correlation functions of the slow variables these asymptotic expressions do not extrapolate back towards the equal time correlations for t going to zero the reason for this initial slip is given and its magnitude calculated method and results are illustrated with a simple example of coupled oscillators

phenomenological criteria for the validity of quantum markovian equations we revisit the constraints on the coefficients of a quantum markovian equation recently obtained by dekker and valsakumar we extend these criteria for time dependent coefficients and establish further criteria where appropriate phenomenological behaviour is required such as dissipation and a boltzmann asymptotic equilibrium state

noncommutative burkholder rosenthal inequalities we investigate martingale inequalities in noncommutative lp spaces associated with a von neumann algebra equipped with a faithful normal state we prove the noncommutative analogue of the classical burkholder inequality on the conditioned or little square function and extend the noncommutative burkholder gundy inequalities from comm math phys 189 1997 667 698 to this nontracial setting we include several related results

irreversibility and randomness in linear response theory recall that the fluctuation dissipation theorem connects the response function of a passive linear system and the spectral density of the stationary stochastic process which describes the thermal fluctuations in the system it is shown that the classical limit 0 of the fluctuation dissipation theorem implies a correspondence between systems which are reversible in the sense that the energy used to drive them away from equilibrium is completely recoverable as work and processes which are deterministic in the sense of wiener s prediction theory while irreversible systems correspond to nondeterministic processes this correspondence is expressed by a simple transformation between the operator kernel which determines the optimal choice of the time dependent force and the linear predictor for the stochastic process for quantum systems this correspondence does not hold the fluctuations are always of the deterministic type for any finite temperature but the system is not necessarily reversible for irreversible systems a formula is derived for the instantaneous entropy production which is a generalization of the standard one for markovian dynamics

calculating two point resistances in distance regular resistor networks an algorithm for the calculation of the resistance between two arbitrary nodes in an arbitrary distance regular resistor network is provided where the calculation is based on stratification introduced in jafarizadeh and salimi 2006 j phys a math gen 39 1 29 and the stieltjes transform of the spectral distribution stieltjes function associated with the network it is shown that the resistances between a node and all nodes belonging to the same stratum with respect to the r i belonging to the ith stratum with respect to the a re the same also the analytical formulae for two point resistances r i i 1 2 3 are given in terms of the size of the network and corresponding intersection numbers in particular the two point resistances in a strongly regular network are given in terms of its parameters v moreover the lower and upper bounds for two point resistances in strongly regular networks are discussed

sufficiency in quantum statistical inference this paper attempts to develop a theory of sufficiency in the setting of non commutative algebras parallel to the ideas in classical mathematical statistics sufficiency of a coarse graining means that all information is extracted about the mutual relation of a given family of states in the paper sufficient coarse grainings are characterized in several equivalent ways and the non commutative analogue of the factorization theorem is obtained as an application we discuss exponential families our factorization theorem also implies two further important results previously known only in finite hilbert space dimension but proved here in generality the koashi imoto theorem on maps leaving a family of states invariant and the characterization of the general form of states in the equality case of strong subadditivity

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

weak limits for quantum random walks we formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions with x n denoting position at time n we show that x n n converges weakly as stackrel ensuremath rightarrow n ensuremath infty to a certain distribution which is absolutely continuous and of bounded support the proof is rigorous and makes use of fourier transform methods this approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods

an elementary construction of lowering and raising operators for the trigonometric calogero sutherland model the quantum calogero sutherland model of an type calogero f 1971 j math phys 12 419 36 sutherland b 1972 phys rev a 4 2019 21 is completely integrable olshanetsky m a and perelomov a m 1977 lett math phys 2 7 13 olshanetsky m a and perelomov a m 1978 funct anal appl 12 121 8 olshanetsky m a and perelomov a m 1983 phys rep 94 313 404 using this fact we give an elementary construction of lowering and raising operators for the trigonometric case this is similar to but more complicated due to the fact that the energy spectrum is not equidistant than the construction for the rational case perelomov a m 1976 itep preprint no 27

on the relationship betweent1 andt2 for stochastic relaxation models we consider the relaxation dynamics of two quantum levels coupled to a stochastic bath we emphasize that even if the matrix elements of the fluctuating hamiltonian are gaussian a second order cumulant truncation is not exact for various stochastic models including the case of a spin 1 2 particle in a fluctuating magnetic field we calculate 1 t1 the population relaxation rate and 1 t2 the phase relaxation rate up to fourth order in perturbation theory we show that unlike the commonly accepted second order result that 1 t2 1 2t1 when fourth order terms are included in some instances 1 t2 1 2t1

markov quantum semigroups admit covariant markov c dilations through a daniell kolmogorov type construction to any markov quantum semigroup on ac algebra there is associated a quantum stochastic process which is a dilation of the semigroup and satisfies a covariance rule which implies the weak markov property

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

free convolution operators and free hall transform abstract we define an extension of the polynomial calculus on a w probability space by introducing an algebra c x i i i which contains polynomials this extension allows us to define transition operators for additive and multiplicative free convolution it also permits us to characterize the free segal bargmann transform and the free hall transform introduced by biane in a manner which is closer to classical definitions finally we use this extension of polynomial calculus to prove two asymptotic results on random matrices the convergence for each fixed time as n tends to of the distribution of the brownian motion on the linear group gl n c to the distribution of a free multiplicative circular brownian motion and the convergence of the classical hall transform on u n to the free hall transform

time ordering techniques and solution of differential difference equation appearing in quantum optics time ordering techniques based on the magnus expansion and the wei norman algebraic procedure are discussed and their relevance and usefulness to quantum optics are stressed

virtues and limitations of markovian master equations with a time dependent generator a markovian master equation with time dependent generator is constructed that respects basic constraints of quantum mechanics in particular the von neumann conditions for the case of a two level system bloch equations with time dependent parameters are obtained necessary conditions on the latter are formulated by employing a time local counterpart of the nakajima zwanzig equation we establish a relation with unitary dynamics we also discuss the relation with the weak coupling limit on the basis of a uniqueness theorem a standard form for the generator of time local master equations is proposed the jaynes cummings model with atomic damping is solved the solution explicitly demonstrates that reduced dynamics can be described by time local master equations only on a finite time interval this limitation is caused by divergencies in the generator a limit of maximum entropy is presented that corroborates the foregoing statements a second limiting case demonstrates that divergencies may even occur for small perturbations of the weak coupling regime

the structure of strongly additive states and markov triplets on the car algebra we find a characterization of states satisfying equality in strong subadditivity of entropy and of markov triplets on the car algebra for even states a more detailed structure of the density matrix is given

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

complete positivity for time dependent qubit master equations it is shown that if the decoherence matrix corresponding to a qubit master equation has a block diagonal real part then the evolution is determined by a one dimensional oscillator equation further when the full decoherence matrix is block diagonal then the necessary and sufficient conditions for completely positive evolution may be formulated in terms of the oscillator hamiltonian or lagrangian when the solution of the oscillator equation is not known an explicit sufficient condition for complete positivity can still be obtained based on a hamiltonian lagrangian inequality a rotational form invariance property is used to characterize the evolution via a single first order nonlinear differential equation enabling some further exact results to be obtained a class of master equations is identified for which complete positivity reduces to the simpler condition of positivity

strict weak mixing of some c dynamical systems based on free shifts abstract we define a stronger property than unique ergodicity with respect to the fixed point subalgebra firstly investigated in b abadie k dykema unique ergodicity of free shifts and some other automorphisms of c algebras math oa 0608227 such a property is denoted as f strict weak mixing f stands for the unital completely positive projection onto the fixed point operator system then we show that the free shifts on the reduced c algebras of rd groups including the free group on infinitely many generators and amalgamated free product c algebras considered in b abadie k dykema unique ergodicity of free shifts and some other automorphisms of c algebras math oa 0608227 are all strictly weak mixing and not only uniquely ergodic