existence positivity and contractivity for quantum stochastic flows with infinite dimensional noise quantum stochastic differential equations of the form govern stochastic flows on a c algebra we analyse this class of equation in which the matrix of fundamental quantum stochastic integrators is infinite dimensional and the coefficient matrix consists of bounded linear operators on weak and strong forms of solution are distinguished and a range of regularity conditions on the mapping matrix are considered for investigating existence and uniqueness of solutions necessary and sufficient conditions on are determined for any sufficiently regular weak solution k to be completely positive the further conditions on for k to also be a contraction process are found and when is a von neumann algebra and the components of are normal these in turn imply sufficient regularity for the equation to have a strong solution weakly multiplicative and homomorphic solutions and their generators are also investigated we then consider the right and left hudson parthasarathy equations in which f is a matrix of bounded hilbert space operators their solutions are interchanged by a time reversal operation on processes the analysis of quantum stochastic flows is applied to obtain characterisations of the generators f of contraction isometry and coisometry processes in particular weak solutions that are contraction processes are shown to have bounded generators and to be necessarily strong solutions

the ito clifford integral abstract we present a theory of non commutative stochastic integration analogous to the ito theory it is shown that wick products of fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted operatorvalued square integrable integrands for square integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined and a doob meyer decomposition for supermartingales obtained

quantum stochastic calculus on full fock modules abstract we develop a quantum stochastic calculus on full fock modules over arbitrary hilbert b b modules we find a calculus of bounded operators where all quantum stochastic integrals are limits of riemann stieltjes sums after having estalished existence and uniqueness of solutions of a large class of quantum stochastic differential equations we find necessary and sufficient conditions for unitarity of a subclass of solutions as an application we find dilations of a conservative cp semigroup quantum dynamical semigroup on b with arbitrary bounded christensen evans generator we point out that in the case b b g the calculus may be interpreted as a calculus on the full fock space tensor initial space g with arbitrary degree of freedom dilating cp semigroups with arbitrary lindblad generator finally we show how a calculus on the boolean fock module reduces to our calculus as a special case this includes a calculus on the boolean fock space

quantum risk sensitive estimation and robustness this paper studies a quantum risk sensitive estimation problem and investigates robustness properties of the filter this is a direct extension to the quantum case of analogous classical results all investigations are based on a discrete approximation model of the quantum system under consideration this allows us to study the problem in a simple mathematical setting we close the paper with some examples that demonstrate the robustness of the risk sensitive estimator

quantum poisson processes and dilations of dynamical semigroups the notion of a quantum poisson process over a quantum measure space is introduced this process is used to construct new quantum markov processes on the matrix algebram n with stationary faithful state if is the quantum measure space in question a von neumann algebra and a faithful normal weight then the semigroupe tl of transition operators on m n has generator whereu is an arbitrary unitary element of the centraliser of m n

temperature states on loop groups boson fermion correspondence and a determinant identity the boson fermion correspondence for the representation of the car algebra over l2 s1 c defined by the kms state with chemical potential is considered and the non bijectivity shown using an alternative formulation the correlations are recalculated leading to a determinant identity reminiscent of szego s theorem

the quantum state diffusion model applied to open systems a model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation derived from a density operator equation such as the bloch equation and consistent with it the advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described the method is applied to the nonlinear absorber cascades of quantum transitions second harmonic generation and a measurement reduction process the model provides graphic illustrations of these processes with statistical fluctuations that mimic those of experiments the stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics the model is compared with the quantum jump model of dalibard 1992 carmichael and others which originated among experimenters looking for intuitive pictures and rules of computation

the quantum theory of measurement within dynamical reduction models we analyse in mathematical detail within the framework of the qmupl model of spontaneous wavefunction collapse the von neumann measurement scheme for the measurement of a 1 2 spin particle we prove that according to the equation of the model i throughout the whole measurement process the pointer of the measuring device is always perfectly well localized in space ii the probabilities for the possible outcomes are distributed in agreement with the born probability rule iii at the end of the measurement the state of the microscopic system has collapsed to the eigenstate corresponding to the measured eigenvalue this analysis shows rigorously how dynamical reduction models provide a consistent solution to the measurement problem of quantum mechanics

detecting symmetries by branch cut the np hard problem of finding symmetries in an abstract graph plays an important role in automatic graph drawing and other applications in this paper we present an exact algorithm for automorphism and symmetry detection based on the branch cut technique we introduce ip models for these problems and investigate the structure of the corresponding polytopes for automorphisms a complete description of the polytope is derived from a given set of generators of the automorphism group the rotation polytopes are shown to be related to the asymmetric traveling salesman polytope while the reflection polytope is related to the matching polytope the algorithm was implemented within the abacus framework and proves to run fast in practice

a lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets given any family of valid inequalities for the asymmetric traveling salesman polytopep g defined on the complete digraphg we show that all members of are facet defining if the primitive members of usually a small subclass are based on this result we then introduce a general procedure for identifying new classes of facet inducing inequalities forp g by lifting inequalities that are facet inducing forp g whereg is some induced subgraph ofg unlike traditional lifting where the lifted coefficients are calculated one by one and their value depends on the lifting sequence our lifting procedure replaces nodes ofg with cliques ofg and uses closed form expressions for calculating the coefficients of the new arcs which are sequence independent we also introduce a new class of facet inducing inequalities the class of sd source destination inequalities which subsumes as special cases most known families of facet defining inequalities

on the monotonization of polyhedra in polyhedral combinatorics one often has to analyze the facial structure of less than full dimensional polyhedra the presence of implicit or explicit equations in the linear system defining such a polyhedron leads to technical difficulties when analyzing its facial structure it is therefore customary to approach the study of such a polytopep through the study of one of its full dimensional relaxations monotonizations known as the submissive and the dominant ofp finding sufficient conditions for an inequality that induces a facet of the submissive or the dominant of a polyhedron to also induce a facet of the polyhedron itself has been posed in the literature as an important research problem our paper goes a long way towards solving this problem we address the problem in the framework of a generalized monotonization of a polyhedronp g mon p that subsumes both the submissive and the dominant and give a sufficient condition for an inequality that defines a facet of g mon p to define a facet ofp for the important cases of the traveling salesman ts polytope in both its symmetric and asymmetric variants and of the linear ordering polytope we give sufficient conditions trivially easy to verify for a facet of the monotone completion to define a facet of the original polytope itself

scattering theory for the belavkin equation describing a quantum particle with continuously observed coordinate in the article the complete investigation is given of the large times behavior of the solutions of the belavkin quantum filtering equation describing a quantum particle with continuously observed coordinate it turns out that these solutions have an extraordinary property namely as t any solution will tend to a gaussian function for almost all realizations of innovating wiener process as a consequence it follows that the dispersion of coordinate will always tend to the same limit as t not depending on initial data

derivation of exact master equation with stochastic description dissipative harmonic oscillator a systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description for the caldeira leggett model of the harmonic oscillator bath a detailed and elementary derivation of the bath induced stochastic field is presented the dynamics of the system is thereby fully described by a stochastic differential equation and the desired master equation would be acquired with statistical averaging it is shown that the existence of a closed form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function for a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled and the master equation naturally results such an equation possesses the lindblad form in which time dependent coefficients are determined by a set of integral equations it is proved that the obtained master equation is equivalent to the well known hu paz zhang equation based on the path integral technique the procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time dependent fields

q deformed poisson random variables on q fock space the q deformed poisson distribution has been introduced as the orthogonalizing probability measure for a certain q deformation of charlier polynomials which is reduced to the usual poisson distribution in the limit q 1 and takes the free poisson distribution in case of q 0 we give the operator on the q fock space which has the q deformed poisson distribution with respect to the vacuum state it is a linear combination of a q number operator a q gaussian random variable and a scalar operator this representation is of the same form as of hudson and parthasarathy on the symmetric fock space

superderivations and symmetric markov semigroups unbounded superderivations are used to construct non commutative elliptic operators on semi finite von neumann algebras the method exploits the interplay between dynamical semigroups and dirichlet forms the elliptic operators may be viewed as generators of irreversible dynamics for fermion systems with infinite degrees of freedom

some aspects of the boson fermion in equivalence a remark on the paper by hudson and parthasarathy for original paper see commun math phys vol 104 p 457 70 1986 the author links the recently proposed unification of the boson and fermion stochastic calculus with the general problem of boson fermion equivalence duality reciprocity etc for quantum fields even if via the fock construction the common fock space for bosons and fermions can be introduced it still does not allow for the unrestricted boson fermion equivalence for field theory models all local fermion field theory models thus have boson equivalents violating the weak local commutativity condition for space dimension three the reverse statement is not valid not all boson models admit a pure fermion reconstruction

representations of lie superalgebras and generalized boson fermion equivalence in quantum stochastic calculus the boson fermion equivalence scheme of 5 can be generalized ton dimensions withr boson andn r fermion creation and annihilation fields the same stochastic integral prescription replaces then 2 generalized number processes ai j which form representations of the lie algebragl n by processes forming representations of the lie superalgebragl n r

the hamiltonian operator associated with some quantum stochastic evolutions we consider the hamiltonian operator associated to the quantum stochastic differential equation introduced by hudson and parthasarathy to describe a quantum mechanical evolution in the presence of a quantum noise we characterize such a hamiltonian in the case of arbitrary multiplicity and bounded coefficients we find an essentially self adjoint restriction of the operator and in particular we provide an explicit construction of a dense set of vectors belonging to its domain

martingale models for quantum state reduction stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory here we examine the correspondence of the two theories in detail making a systematic use of the methods of martingale theory an analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred we consider the general projection postulate of l uders applicable in the case of a possibly degenerate eigenvalue spectrum and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state we also analyse the associated lindblad equation for the evolution of the density matrix and obtain an exact time dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the l uders density matrix finally we apply girsanov s theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric brownian motions thereby constructing an explicit unravelling of the lindblad equation

nonperturbative treatment of non markovian dynamics of open quantum systems we identify the conditions that guarantee equivalence of the reduced dynamics of an open quantum system oqs for two different types of environments one a continuous bosonic environment leading to a unitary system environment evolution and the other a discrete mode bosonic environment resulting in a system mode nonunitary lindbladian evolution assuming initial gaussian states for the environments we prove that the two oqs dynamics are equivalent if both the expectation values and two time correlation functions of the environmental interaction operators are the same at all times for the two configurations since the numerical and analytical description of a discrete mode environment undergoing a lindbladian evolution is significantly more efficient than that of a continuous bosonic environment in a unitary evolution our result represents a powerful nonperturbative tool to describe complex and possibly highly non markovian dynamics as a special application we recover and generalize the well known pseudomodes approach to open system dynamics

meixner class of non commutative generalized stochastic processes with freely independent values ii the generating function let t be an underlying space with a non atomic measure on it in comm math phys 292 99 129 2009 the meixner class of non commutative generalized stochastic processes with freely independent values omega omega t t in t was characterized through the continuity of the corresponding orthogonal polynomials in this paper we derive a generating function for these orthogonal polynomials the first question we have to answer is what should serve as a generating function for a system of polynomials of infinitely many non commuting variables we construct a class of operator valued functions z z t t in t such that z t commutes with s for any s t in t then a generating function can be understood as g z omega sum n 0 infty int t n p n omega t 1 dots omega t n z t 1 dots z t n sigma dt 1 dots sigma dt n where p n omega t 1 dots omega t n is the kernel of the nth orthogonal polynomial we derive an explicit form of g z which has a resolvent form and resembles the generating function in the classical case albeit it involves integrals of non commuting operators we finally discuss a related problem of the action of the annihilation operators partial t t in t in contrast to the classical case we prove that the operators t related to the free gaussian and poisson processes have a property of globality this result is genuinely infinite dimensional since in one dimension one loses the notion of globality

the ito algebra of quantum gaussian fields abstract the notion of mutual quadratic variation square bracket is extended to a quantum probabilistic framework the mutual quadratic variations of the annihilation creation and number fields in a gaussian representation are calculated in both the boson and the fermion case in the strong topology on a common invariant domain it is proved that the corresponding ito table closes at the second order the fock representation is characterized among the gaussian ones by the property that its ito table closes at the first order

stochastic integral representation of bounded quantum martingales in fock space it is a classical theorem of kunita and watanabe 6 that every square integrable martingale adapted to the standard brownian motion can be uniquely expressed as the stochastic integral of a nonanticipating square integrable brownian functional with respect to the same brownian motion the central aim of this paper is to establish a similar integral represen tation of a quantum martingale with respect to the annihilation creation and gauge processes in the context of quantum stochastic calculus in fock space as developed in 11 in the case of non fock quantum brownian motion such an integral representation was achieved by hudson and lindsay 2 33 with much less difficulty owing to the non existence of the so called gauge processes as special cases of our main result we obtain the differentials of hilbert schmidt and unitary martingales 4 as an application the uniqueness of fermion martingales in boson fock space is established the first author wishes to thank hudson and lindsay for several useful conversations on this subject 2

continuous quantum jumps and infinite dimensional stochastic equations from a mathematical point of view a class of infinite dimensional stochastic differential equations describing continuous spontaneous localization in quantum dynamics will be studied existence and uniqueness of weak and strong solutions of respective equations are proven via cameron martin girsanov transformation the case of gaussian initial states is explicitly solved

solving quantum stochastic differential equations with unbounded coefficients we demonstrate a method for obtaining strong solutions to the right hudson parthasarathy quantum stochastic differential equation dut f utd t u0 1 where u is a contraction operator process and the matrix of coefficients f consists of unbounded operators this is achieved whenever there is a positive self adjoint reference operator c that behaves well with respect to the f allowing us to prove that domc1 2 is left invariant by the operators ut thereby giving rigorous meaning to the formal expression above we give conditions under which the solution u is an isometry or coisometry process and apply these results to construct unital homomorphic dilations of quantum markov semigroups arising in probability and physics

facet identification for the symmetric traveling salesman polytope several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given an identification procedure accepts as input the support graph of a point which does not belong to the polytope and returns as output some of the facet inducing inequalities violated by the point a procedure which always accomplishes this task is calledexact otherwise it is calledheuristic we give exact procedures for the subtour elimination and the 2 matching constraints based on the gomory hu and padberg rao algorithms respectively efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities these procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large scale problem instances with sizes up to 2392 nodes to optimality

on martingale inequalities in non commutative stochastic analysis we develop a non commutativelpstochastic calculus for the clifford stochastic integral anl2theory of which has been developed by barnett streater and wilde the main results are certain non commutativelpinequalities relating clifford integrals and their integrands these results are applied to extend the domain of the clifford integral froml2tol1integrands and we give applications to optional stopping of clifford martingales proving an analog of a theorem of burkholder the stopped clifford processfthas zero expectation providedet in proving these results we establish a number of results relating the clifford integral to the differential calculus in the clifford algebra in particular we show that the clifford integral is given by the divergence operator and we prove an explicit martingale representation theorem both of these results correspond closely to basic results for stochastic analysis on wiener space thus furthering the analogy between the clifford process and brownian motion

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

a martingale characterization of quantum poisson processes we study the characterization problem of quantum poisson processes using the martingale property and some conditions on moments up to fourth order these conditions are inspired by an assumption on the quadratic variation that is equivalent in classical cases to the fact that the process has purely discontinuous sample paths with unit jumps

solving the asymmetric travelling salesman problem with time windows by branch and cut many optimization problems have several equivalent mathematical models it is often not apparent which of these models is most suitable for practical computation in particular when a certain application with a specific range of instance sizes is in focus our paper addresses the asymmetric travelling salesman problem with time windows atsp tw from such a point of view the real world application we aim at is the control of a stacker crane in a warehouse we have implemented codes based on three alternative integer programming formulations of the atsp tw and more than ten heuristics computational results for real world instances with up to 233 nodes are reported showing that a new model presented in a companion paper outperforms the other two models we considered at least for our special application and that the heuristics provide acceptable solutions

quantum ω semimartingales and stochastic evolutions we explore adaptedness a variant of the usual notion of adaptedness found in stochastic calculus it is shown that the non adapted quantum stochastic integrals of bounded adapted processes are themselves bounded and adapted a fact that may be deduced from the bismut clark ocone formula of malliavin calculus an algebra analogous to attal s class of regular quantum semimartingales is defined and product and functional ito formulae are given we consider quantum stochastic differential equations with bounded adapted coefficients that are time dependent and act on the whole fock space solutions to such equations may be used to dilate quantum dynamical semigroups in a manner that generalises and gives new insight into that of r alicki and m fannes 1987 comm math phys 108 353 361 their unitarity condition is seen to be the usual condition of r l hudson and k r parthasarathy 1984 comm math phys93 301 323

a stochastic stinespring theorem completely positive markovian cocycles on a von neumann algebra adapted to a fock filtration are realised as conjugations of homomorphic markovian cocycles the conjugating processes are affiliated to the algebra and are governed by quantum stochastic differential equations whose coefficients evolve according to the homomorphic process some perturbation theory for quantum stochastic flows is developed in order to achieve the above stinespring decomposition

the series product and its application to quantum feedforward and feedback networks the purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks these methods build on and generalize existing methods for series or cascade connections by allowing for more general interfaces and by introducing an efficient algebraic tool the series product we also introduce another product which we call the concatenation product that is useful for assembling and representing systems without necessarily having connections we show how the concatenation and series products can be used to describe feedforward and feedback networks a selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology

coherent quantum lqg control based on a recently developed notion of physical realizability for quantum linear stochastic systems we formulate a quantum lqg optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum such a control scheme is often referred to in the quantum control literature as coherent feedback control it distinguishes the present work from previous works on the quantum lqg problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom the difference in our formulation is the presence of additional non linear and linear constraints on the coefficients of the sought after controller rendering the problem as a type of constrained controller design problem due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard lqg problem we propose a numerical procedure for solving this problem based on an alternating projections algorithm and as an initial demonstration of the feasibility of this approach we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained for comparison we also consider the case of classical linear controllers that use direct or indirect measurements and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones

the precedence constrained asymmetric traveling salesman polytope many applications of the traveling salesman problem require the introduction of additional constraints one of the most frequently occurring classes of such constraints are those requiring that certain cities be visited before others precedence constraints in this paper we study the precedence constrained asymmetric traveling salesman pcats polytope i e the convex hull of incidence vectors of tours in a precedence constrained directed graph we derive several families of valid inequalities and give polynomial time separation algorithms for important subfamilies we then establish the dimension of the pcats polytope and show that under reasonable assumptions the two main classes of inequalities derived are facet inducing

some self adjoint quantum semimartingales it is proved that the quantum stochastic gauge integral preserves self adjointness of vacuum adapted processes this fact together with bounded perturbations and the link between the hudson parthasarathy calculus and vacuum adapted theory is used to produce many self adjoint quantum semimartingales

on dissipative stochastic equations in a hilbert space a general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a hilbert space is proved the dual equation is introduced and the duality relation is established proofs take inspirations from quantum stochastic calculus however without using it solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup describing quantum markovian evolution the problem of the mean square norm conservation closely related to the unitality non explosion of the quantum dynamical semigroup is considered and a hyperdissipativity condition ensuring such conservation is discussed comments are given on the existence of solutions of a nonlinear stochastic differential equation introduced and discussed recently in physical literature in connection with continuous quantum measurement processes

the essentially commutative dilations of dynamical semigroups on m n for identity and trace preserving one parameter semigroups tt t 0 on then n matricesmn we obtain a complete description of their essentially commutative dilations i e dilations which can be constructed on a tensor product ofmn by a commutativew algebra

facets of the asymmetric traveling salesman polytope in this paper we consider the asymmetric traveling salesman polytope pn defined as the convex hull of the incidence vectors of tours in a complete digraph with n vertices and its monotonization pi n several classes of valid inequalities for both there polytopes have been introduced in the literature which are not known to define facets of pn or pi n we describe a general technique which can often be used to prove that a given inequality defines a facet of pn the method is applied to prove that the so called dk dk c3 comb and c2 inequalities define facets of pn thus solving open problems as for the monotone polytope a necessary and sufficient condition for a facet defining inequality of pn to define a facet of pi n as well is introduced which applies to the dk dk c3 comb and c2 inequalities in addition a simple procedure for transforming any facet defining inequality of pn into an equivalent one which also defines a facet of pi n is given

a representation free quantum stochastic calculus abstract we develop a representation free stochastic calculus based on three inequalities semimartingale inequality scalar forward derivative inequality scalar conditional variance inequality we prove that our scheme includes all the previously developed stochastic calculi and some new examples the abstract theory is applied to prove a boson levy martingale representation theorem in bounded form and a general existence uniqueness and unitarity theorem for quantum stochastic differential equations

noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations the notion of a unitary noncommutative stochastic process with independent and stationary increments is introduced and it is proved that such a process under a continuity assumption can be embedded into the solution of a quantum stochastic differential equation in the sense of hudson and parthasarathy 8

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

mayer problem for quantum stochastic control this paper is concerned with the value function for optimal quantum stochastic control we established the lipschitz continuity of the value function which is inherent from the lipschitz property of the set valued map describing the quantum stochastic control by using a feedback multivalued map we also prove a characterization of optimal solutions relating to a noncommutative generalization of mayer problem

localization of quantum bernoulli noises the family k k k 0 of annihilation and creation operators acting on square integrable functionals of a bernoulli process z zk k 0 can be interpreted as quantum bernoulli noises in this note we consider the operator family lk lk k 0 where lk kek with ek being the conditional expectation operator given field zj 0 j k we show that lk resp lk is essentially a kind of localization of the annihilation operator k resp creation operator k we examine properties of the family lk lk k 0 and prove among other things that lk and lk satisfy a local canonical anti communication relation and lk k 0 forms a mutually orthogonal operator sequence although each lk is not a projection operator we find that the operator series k 0 lk xlk converges in the strong operator topology for each bounded operator x acting on square integrable functionals of z in particular we get an explicit sum of the operator series k 0 lk lk a useful norm estimate on k 0 lk xlk is also obtained final

a representation of the belavkin equation via feynman path integrals the belavkin equation describing the continuous measurement of the position of a quantum particle is studied a rigorous representation of its solution by means of an infinite dimensional oscillatory integral feynman path integral defined on the complex cameron martin space is given

distributed regression learning with coefficient regularization abstract we study distributed regression learning with coefficient regularization scheme in a reproducing kernel hilbert space rkhs the algorithm randomly partitions the sample set z i i 1 n into m disjoint sample subsets of equal size applies the coefficient regularization scheme to each sample subset to produce an output function and averages the individual output functions to get the final global estimator we deduce the error bound in expectation in the l 2 metric and prove the asymptotic convergence for this distributed coefficient regularization learning satisfactory learning rates are then derived under a standard regularity condition on the regression function which reveals an interesting phenomenon that when m n s and s is small enough this distributed learning has the same convergence rate compared with the algorithm processing the whole data in one single machine

lifted cycle inequalities for the asymmetric traveling salesman problem we investigate the family of facet defining inequalities for the asymmetric traveling salesman ats polytope obtainable by lifting the cycle inequalities we establish several properties of this family that earmark it as arguably the most important among the asymmetric inequalities for the ats polytope known to date in particular we give several results that characterize the pattern of lifting coefficients and thereby facilitate the identification of facet defining inequalities this characterization is used to investigate three new large classes of lifted cycle inequalities that we call shell fork and curtain inequalities these latter inequalities have unbounded chvatal rank furthermore the pattern of lifting coefficients that we identify makes it easy to develop efficient separation routines finally each member of the family is shown to have a counterpart for the symmetric ts sts polytope that is often new and is obtainable by mapping the inequality for the ats polytope into a certain face of the sts polytope and then lifting the resulting inequality into one for the sts polytope itself

quantum q white noise and a q central limit theorem we establish a connection between the azema martingales and certain quantum stochastic processes with increments satisfyingq commutation relations this leads to a theory ofq white noise onq bialgebras and to a generalization of the fock space representation theorem for white noise on bialgebras in particular quantum azema noise q interpolations between fermion and boson quantum brownian motion and unitary evolutions withq independent multiplicative increments are studied it follows from our results that the azema martingales and theq interpolations are central limits of sums ofq independent identically distributed quantum random variables

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

singular perturbation approximations for a class of linear quantum systems this technical note considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics the technical note presents results on the physical realizability properties of the approximate system arising from singular perturbation model reduction

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

noncommutative brownian motion in monotone fock space an example of noncommutative brownian motion is constructed on the monotone fock space which is a kind of fock space generated by all the decreasing finite sequences of positive real numbers the probability distribution at time associated to this brownian motion is shown to be the arcsine law normalized to mean 0 and variance t

a martingale characterization of canonical commutation and anticommutation relations using a martingale condition and some restrictions on moments up to fourth order the characterisation problem of boson fermion and classical brownian motions is studied from a unified point of view entirely within the framework of elementary operator theory global commutation and anticommutation rules turn out to be consequences of corresponding commutation and anticommutation rules between past and future observables

diagonal versions and quantum stochastic integrals on the symmetric fock space with nonadapted integrands we develop a quantum stochastic calculus for nonadapted processes and the three standard integrators on the symmetric fock space over an arbitrary polish space the integrable processes are restricted to a class of operators which possess some kernel description our approach is mainly based on the concept of diagonalized versions 16 in the case of the real line we consider connections to the usual stochastic and quantum stochastic calculi and derive ito formulae besides the standard integrals with two integrands and one integrator these formulae lead to a new type of quantum stochastic integrals which has two integrators and three integrands

debiased magnitude preserving ranking learning rate and bias characterization abstract magnitude preserving ranking mprank under tikhonov regularization framework has shown competitive performance on information retrial besides theoretical advantages for computation feasibility and statistical guarantees in this paper we further characterize the learning rate and asymptotic bias of mprank and then propose a new debiased ranking algorithm in terms of the operator representation and approximation techniques we establish their convergence rates and bias characterizations these theoretical results demonstrate that the new model has smaller asymptotic bias than mprank and can achieve the satisfactory convergence rate under appropriate conditions in addition some empirical examples are provided to verify the effectiveness of our debiased strategy

continuous unital dilations of completely positive semigroups abstract since it began in the 1970s the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup several authors have proved the existence of dilations but have usually only achieved unital dilations of specific examples while general theorems have tended to produce non unital dilations a unique approach due to jean luc sauvageot leads to a unital dilation in a general setting but leaves unclear the continuity of the dilation semigroup the major purpose of this paper therefore is to further develop sauvageot s theory in order to prove the existence of continuous unital dilations the dilation is characterized by a combinatorial property akin to free independence this property is implicit in some of sauvageot s original calculations but a secondary goal of this paper is to present it as its own object of study which is done in section 4 this work is based on the author s ph d thesis

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

on the energy increase in space collapse models a typical feature of spontaneous collapse models which aim at localizing wavefunctions in space is the violation of the principle of energy conservation in the models proposed in the literature the stochastic field which is responsible for the localization mechanism causes the momentum to behave like a brownian motion whose larger and larger fluctuations show up as a steady increase of the energy of the system in spite of the fact that in all situations such an increase is small and practically undetectable it is an undesirable feature that the energy of physical systems is not conserved but increases constantly in time diverging for t in this paper we show that this property of collapse models can be modified we propose a model of spontaneous wavefunction collapse sharing all most important features of usual models but such that the energy of isolated systems reaches an asymptotic finite value instead of increasing with a steady rate

spectral families of quantum stochastic integrals we develop a theory of spectral integration for quantum stochastic integrals of certain families of processes driven by creation conservation and annihilation processes in fock space these give a non commutative generalisation of classical stochastic integrals driven by poisson random measures a stochastic calculus for these processes is developed and used to obtain unitary operator valued solutions of stochastic differential equations as an application we construct stochastic flows on operator algebras driven by levy processes with finite levy measure

filtered random variables bialgebras and convolutions we introduce the filtered bialgebra which is a multivariate generalization of the unital bialgebra c x x p of polynomials in noncommuting variables x x x x and a projection p p p2 endowed with the coproduct x x 1 1 x x x p p x with p being group like we study the associated convolutions random walks and filtered random variables the gns representations of the limit states lead to filtered fundamental operators which are the ccr fundamental operators on the multiple symmetric fock space h over h l2 r g where g is a separable hilbert space multiplied by appropriate projections the importance of filtered random variables and fundamental operators stems from the fact that by addition and strong limits one obtains from them the main types of noncommutative random variables and fundamental operators respectively regardless of the type of noncommutative independence

how stands collapse i in this volume in honour of giancarlo ghirardi i discuss my involvement with ideas of dynamical collapse of the state vector ten problems are introduced nine of which were seen following my initial work four of these problems had a resolution in giancarlo ghirardi alberto rimini and tullio weber s spontaneous localization sl model which added one more problem this stimulated a somewhat different resolution of these five problems in the continuous spontaneous localization csl model in which i combined my initial work with sl in an upcoming volume in honour of abner shimony i shall discuss the status of the remaining five post csl problems

towards quantum superpositions of a mirror an exact open systems analysis calculational details we give details of calculations analysing the proposed mirror superposition experiment of marshall simon penrose and bouwmeester within different stochastic models for state vector collapse we give two methods for exactly calculating the fringe visibility in these models one proceeding directly from the equation of motion for the expectation of the density matrix and the other proceeding from solving a linear stochastic unravelling of this equation we also give details of the calculation that identifies the stochasticity parameter implied by the small displacement taylor expansion of the csl model density matrix equation the implications of the two results are briefly discussed two pedagogical appendices review mathematical apparatus needed for the calculations

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

the fixed outdegree 1 arborescence polytope a 1 arborescence is a spanning arborescence rooted at node 1 plus one arc incident into node 1 the 1 arborescence polytope the convex hull of incidence vectors of 1 arborescences is a well known relaxation of the asymmetric traveling salesman ats polytope in this paper we study the fixed outdegree 1 arborescence fda polytope defined as the convex hull of incidence vectors of 1 arborescences having outdegree 1 at a prescribed subset f of nodes the fda polytope becomes the 1 arborescence polytope when f and it becomes the ats polytope when f is the entire node set we derive several classes of valid inequalities for the fda polytope and show that they are facet defining this is done by using a theorem that establishes sufficient conditions for a facet defining inequality for the ats polytope to also be facet defining for the fda polytope we then introduce the class of fd inequalities and show that they define facets of both the ats polytope and the fda polytope finally we define a canonical form that facilitates rigorous comparisons between inequalities and use it to show that the new facets are distinct from each other and from other known facets

collapse models with non white noises we set up a general formalism for models of spontaneous wavefunction collapse with dynamics represented by a stochastic differential equation driven by general gaussian noises not necessarily white in time in particular we show that the non schrodinger terms of the equation induce the collapse of the wavefunction to one of the common eigenstates of the collapsing operators and that the collapse occurs with the correct quantum probabilities we also develop a perturbation expansion of the solution of the equation with respect to the parameter which sets the strength of the collapse process such an approximation allows one to compute the leading order terms for the deviations of the predictions of collapse models with respect to those of standard quantum mechanics this analysis shows that to leading order the imaginary noise trick can be used for non white gaussian noise