an elementary proof of the strong law of large numbers in the following note we present a proof for the strong law of large numbers which is not only elementary in the sense that it does not use kolmogorov s inequality but it is also more applicable because we only require the random variables to be pairwise independent an extension to separable banach space valuedr dimensional arrays of random vectors is also discussed for the weak law of large numbers concerning pairwise independent random variables which follows from our result see theorem 5 2 2 in chung 1

multichoice minority game the generalization of the problem of adaptive competition known as the minority game to the case of k possible choices for each player is addressed and applied to a system of interacting perceptrons with input and output units of a type of k state potts spins an optimal solution of this minority game as well as the dynamic evolution of the adaptive strategies of the players are solved analytically for a general k and compared with numerical simulations

sliding mode control of two level quantum systems this paper proposes a robust control method based on sliding mode design for two level quantum systems with bounded uncertainties an eigenstate of the two level quantum system is identified as a sliding mode the objective is to design a control law to steer the system s state into the sliding mode domain and then maintain it in that domain when bounded uncertainties exist in the system hamiltonian we propose a controller design method using the lyapunov methodology and periodic projective measurements in particular we give conditions for designing such a control law which can guarantee the desired robustness in the presence of the uncertainties the sliding mode control method has potential applications to quantum information processing with uncertainties

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

an extreme value theory for long head runs for an infinite sequence of independent coin tosses withp heads p 0 1 the longest run of consecutive heads in the firstn tosses is a natural object of study we show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn 1 p i i d exponential random variables these results are extended to the case of the longest head run interrupted byk tails the mean length of this run is shown to be log n klog n k 1 log 1 p log k k 1 2 r1 n o 1 where log log1 p 0 577 is the euler mascheroni constant ln 1 p andr1 n is small the variance is 2 6 2 1 12 r2 n o 1 wherer2 n is again small upper and lower class results for these run lengths are also obtained and extensions discussed

invariance principles for renewal processes on presente une methodologie generale pour demontrer des principes d invariance pour des processus de renouvellement on obtient des approximations presque sures et d inegalites de probabilites

the hausdorff dimension of level sets described by erdos renyi average abstract let 0 1 and be an integer function defined on n 0 satisfying 1 n n define the level set e r x 0 1 lim n a n n x where a n n x is the n n erdos renyi average of x 0 1 in this paper we will give descriptions for the hausdorff dimension of e r under the assumption n as n which complement simultaneously an early classic result of besicovitch and the new strong law of large number established by p erdos and a renyi moreover for the case n m ultimately where m 1 is an integer the hausdorff dimension of e r is also determined by us in the last section

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

a discrete invitation to quantum filtering and feedback control the engineering and control of devices at the quantum mechanical level such as those consisting of small numbers of atoms and photons is a delicate business the fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise making this a novel field of application for stochastic estimation and control theory in this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance the model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite dimensional hilbert spaces thus avoiding the technical complications of the continuous theory we introduce discretized models of an atom in interaction with the electromagnetic field obtain filtering equations for photon counting and homodyne detection and solve a stochastic control problem using dynamic programming and lyapunov function methods

quantum measurement based feedback control a nonsmooth time delay control approach this paper addresses the problem of measurement based feedback control for quantum systems in which the time to compute the filter based control input is taken into account by considering the input delay it starts with a new lyapunov lasalle like theorem for delay dependent stochastic stability of a class of time delay stochastic nonlinear systems nonsmooth time delay control is then constructed to compensate for the control computation time which is known but arbitrarily long while globally stabilizing the quantum filters almost surely the nonsmooth property enables the control to deal with the symmetric topology of filter state space the effectiveness of the proposed control is illustrated through the global stabilization of the spin 1 2 systems simulation results are presented and discussed to show the effectiveness of the proposed control

limit laws of erdos renyi shepp type on obtient la forme complete des theoremes d erdos renyi 1970 et shepp 1964 les conditions etendent celles de deheuvels devroye et lynch 1986 pour couvrir une classe de distributions plus grande

non markovian quantum feedback networks i quantum transmission lines lossless bounded real property and limit markovian channels the purpose of this paper is to set out the problems of modeling quantum communication and signal processing where the communication between systems is via a non markovian channel this is a general feature of quantum transmission lines our ultimate objective is to extend the network rules that have been developed for markovian models to this end we recall the hamiltonian description of such non markov models of transmission lines and their quantization these have occurred in the context of non equilibrium thermodynamics but our interest is in the transmission lines as carriers of information rather than heat baths we show that there is an analytic scattering matrix associated with these models and that stability may be formulated in terms of the lossless bounded real property noting that the input and output fields do not separately satisfy a non self demolition principle we discuss the rigorous limit in which such models appear markov and so amenable to standard approaches of quantum filtering an

on the increments of partial sum processes with multidimensional indices consider a set xn nd of iid random variables indexed by nd the positive integer d dimensional lattice points the erdos renyi 1970 law of large numbers is generalized to this case and some extensions are given the results include a theorem of chan 1976 for the d parameter brownian motion

quantum pontryagin principle under continuous measurements in this paper we develop the theory of the quantum pontryagin principle for continuous measurements and feedback the analysis is carried out under the assumption of compatible events in the output channel the plant is a quantum system which generally is in a mixed state coupled to a continuous measurement channel the pontryagin maximum principle is derived in both the schrodinger picture and heisenberg picture in particular in statistical moment coordinates to avoid solving stochastic equations we derive a linear quadratic gaussian scheme which is more suitable for control purposes finally we use the quantum harmonic oscillator as a concrete example to illustrate the performance of the controller

stabilizing feedback controls for quantum systems no quantum measurement can give full information on the state of a quantum system hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory here we study the properties of controlled quantum filtering equations as classical stochastic differential equations we then develop methods using a combination of geometric control and classical probabilistic techniques for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator

limit results on finite dimensional gaussian random fields in this paper we establish some limit results on csorgo revesz type increments combined with moduli of continuity for finite dimensional gaussian random fields under explicit conditions via estimating upper bounds of large deviation probabilities on suprema of the finite dimensional gaussian random fields

fault tolerant control of linear quantum stochastic systems in quantum engineering faults may occur in a quantum control system which will cause the quantum control system unstable or deteriorate other relevant performance of the system this note presents an estimator based fault tolerant control design approach for a class of linear quantum stochastic systems subject to fault signals in this approach the fault signals and some commutative components of the quantum system observables are estimated and a fault tolerant controller is designed to compensate the effect of the fault signals numerical procedures are developed for controller design and an example is presented to demonstrate the proposed design approach

information geometry and local asymptotic normality for multi parameter estimation of quantum markov dynamics this paper deals with the problem of identifying and estimating dynamical parameters of continuous time markovian quantum open systems in the input output formalism first we characterise the space of identifiable parameters for ergodic dynamics assuming full access to the output state for arbitrarily long times and show that the equivalence classes of undistinguishable parameters are orbits of a lie group acting on the space of dynamical parameters second we define an information geometric structure on this space including a principal bundle given by the action of the group as well as a compatible connection and a riemannian metric based on the quantum fisher information of the output we compute the metric explicitly in terms of the markov covariance of certain fluctuation operators and relate it to the horizontal bundle of the connection third we show that the system output and reduced output state satisfy local asymptotic normality i e they can be approximated by a gaussian model consisting of coherent states of a multimode continuos variables system constructed from the markov covariance data we illustrate the result by working out the details of the information geometry of a physically relevant two level system

exact convergence rate in the limit theorems of erdos renyi and shepp on considere le comportement asymptotique de u n sup o i n k s i k s i et on montre que le terme o k 1 2 peut etre remplace par o k 1 log k

lattice point problems and distribution of values of quadratic forms for d dimensional irrational ellipsoids e with d 9 we show that the number of lattice points in re is approximated by the volume of re as r tends to infinity up to an error of order o rd 2 the estimate refines an earlier authors bound of order o rd 2 which holds for arbitrary ellipsoids and is optimal for rational ellipsoids as an application we prove a conjecture of davenport and lewis that the gaps between successive values say s 0 as s oo for d 9 for comparison note that sups n s s 0 for rational q x and d 5 as a corollary we derive oppenheim s conjecture for indefinite irrational quadratic forms i e the set q zd is dense in r for d 9 which was proved for d 3 by g margulis marl in 1986 using other methods finally we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums

continuous mode multiphoton filtering the purpose of this paper is to derive filters for an arbitrary open quantum system driven by a light wave packet prepared in a continuous mode multiphoton state a continuous mode multiphoton stat

fault tolerant quantum filtering and fault detection for quantum systems this paper aims to determine the fault tolerant quantum filter and fault detection equation for a class of open quantum systems coupled to a laser field that is subject to stochastic faults in order to analyze this class of open quantum systems we propose a quantum classical bayesian inference method based on the definition of a so called quantum classical conditional expectation it is shown that the proposed bayesian inference approach provides a convenient tool to simultaneously derive the fault tolerant quantum filter and the fault detection equation for this class of open quantum systems an example of two level open quantum systems subject to poisson type faults is presented to illustrate the proposed method these results have the potential to lead to a new fault tolerant control theory for quantum systems

on the erdos renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings we prove in this paper a law of erdos renyi type for arrays of independent and identically distributed random variables the relationships of our theorem with similar results obtained in the theory of runs and spacings are investigated applications include the evaluation of the rate of convergence of erdos renyi maxima in limiting cases and a generalization of a theorem of erdos and revesz on runs

strong bounds for multidimensional spacings let u1 u2 u n be independent random vectors uniformly distributed on 0 1 d we define the k th maximal spacing n k associated to u1 u1 as the k th maximal possible value of the length of a side of a d dimensional square block in 0 1 d which does not intersect the sample and which cannot be enlarged without doing so our main result is that n k n 1 log n 0 log2n 1 d almost surely as n other bounds are proposed for the limiting almost sure behavior of n k as n

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

functional erdős renyi laws for semiexponential random variables for an i i d sequence of random variables with a semiexponential distribution we give a functional form of the erdos renyi law for partial sums in contrast to the classical case that is the case where the random variables have exponential moments of all orders the set of limit points is not a subset of the continuous functions this reflects the bigger influence of extreme values the proof is based on a large deviation principle for the trajectories of the corresponding random walk the normalization in this large deviation principle differs from the usual normalization and depends on the tail of the distribution in the same way we prove a functional limit law for moving averages

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

quantum optical realization of classical linear stochastic systems abstract the purpose of this paper is to show how a class of classical linear stochastic systems can be physically implemented using quantum optical components quantum optical systems typically have much higher bandwidth than electronic devices meaning faster response and processing times and hence have the potential for providing better performance than classical systems a procedure is provided for constructing the quantum optical realization the paper also describes the use of the quantum optical realization in a measurement feedback loop some examples are given to illustrate the application of the main results

non markovian quantum repeated interactions and measurements a non markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented by adapting and applying usual projection operator techniques in this context discrete versions of the integro differential and time convolutionless master equations for the reduced system are derived next an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non markovian stochastic master equation for the open system is obtained finally the question of unravelling in a particular model of non markovian quantum interactions is discussed

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

limit theorems for moving averages of independent random vectors let xj be i i d r v s taking values in a separable banach space and let s n sum limits j 1 n x j the almost sure limiting behavior of the normalized random set of increments bn 1 sk h sk 1 k n 1 h an for a broad class of sequences an with corresponding normalizations bnis determined when an log n the results generalize and improve some recent work of csorg and revesz and some previous work of lai when an clog n the results generalize and refine in a certain sense the erdos renyi law of large numbers the case an logn 0 is also studied

intrinsic stochastic differential equations as jets we explain how ito stochastic differential equations sdes on manifolds may be defined using 2 jets of smooth functions we show how this relationship can be interpreted in terms of a convergent numerical scheme we also show how jets can be used to derive graphical representations of ito sdes and we show how jets can be used to derive the differential operators associated with sdes in a coordinate free manner we relate jets to vector flows giving a geometric interpretation of the ito stratonovich transformation we show how percentiles can be used to give an alternative coordinate free interpretation of the coefficients of one dimensional sdes we relate this to the jet approach this allows us to interpret the coefficients of sdes in terms of fan diagrams in particular the median of an sde solution is associated with the drift of the sde in stratonovich form for small times

is measurement based feedback still better for quantum control systems in this paper we put forward a fundamental question concerning feedback control of quantum systems is measurement based feedback control still better than open loop control in contrast to the classical control theory the answer is far from obvious this is because measurement based feedback needs measurement to reduce the system uncertainty whereas the measurement on a quantum system will inevitably increase the system uncertainty in turn in fact there is a complicated tradeoff between the uncertainty introduced and the information gained by the measurement on a quantum system to investigate this fundamental problem we will only focus on a typical model of coherent control mode with and without the decoherence term in the paper by establishing some fundamental limits on the performances of both the open loop and measurement based feedback controls we will demonstrate via simulation that the measurement based feedback control of quantum systems is still superior to the open loop control in some sense for the typical model under consideration

strong laws for the maximal k spacing when k c log n we consider the maximal k spacing m n mathop max limits 0 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant n 1 k left u ni k u ni right where u n1 u nn are the order statistics of an i i d sample of size n from the uniform distribution on 0 1 and u n0 0 u nn 1 1 the integer k is allowed to vary with n at a rate not exceeding log n we obtain laws of the iterated logarithm for all the k s in the given range for small k the methods used in the proofs are borrowed from extreme value theory for larger k the techniques are reminiscent of those used in the proof of the erdos renyi theorem

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

large deviations and maximum entropy principle for interacting random fields on mathbb z d we present a new approach to the principle of large deviations for the empirical field of a gibbsian random field on the integer lattice zd this approach has two main features first we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities and thus obtain estimates which are finer and more widely applicable second we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field the limits being those predicted by the maximum entropy principle this result implies a general version of the equivalence of gibbs ensembles stating that every microcanonical limiting state is a grand canonical equilibrium state we also prove a converse to the last statement and discuss some applications 0 introduction as is well known the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles the principle of large deviations and the maximum entropy principle the former provides the exact rate of exponential decay of the fluctuation probabilities whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large it is obvious that these principles are intimately related in this paper we investigate these principles in the case of interacting random fields on the integer lattice 7zd the setup is the following first we let e 4 be any measurable space we shall assume throughout that e a is standard borel but we shall avoid making any explicit topological assumptions on e so we do not assume that e is polish next we let s zd be the d dimensional integer lattice and q f e gas the associated product space in lattice models of statistical mechanics e is called the state space or single spin space and q f the configuration space we let j s f denote the set of all probability

on the increments of the wiener process let w t t 0 be a standard wiener process and 0 a t t a nondecreasing function of t the asymptotic behaviour of several increment processes obtained from w and a t is investigated in terms of their upper classes in some cases we characterize these classes by means of an integral test two such processes are w t a t w t a t 1 2 and mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbf9f4rqqrpu0dxde9lqfhe9lq pg0di9veea0 yr0ryxir jbba9q8aq0 yq he9q8qqi frfve9fve9 ff0dmeaabaqacigacagaaeqabawaaeaaeaaakeaadawfqaqaaigaco hacagg1bgaaiicaawcbaacbagaa83bamrr1ngbprwthrhayaqeguud jxwakbsthrhagq1dvbacfagae4nziyoae4ha3thcbigaa0hdaiaa b cacqgfmjihcqgfgaaiiewacaafubaabeaakiaa bcadawfqaqaaiga cohacagg1bgaaiicaawcbagaa83baiaa bcatcvaufkttlearcwql5 gd1jxegm1yo92bsj2bvtgaiyaacqwe8aapcqgfacpdcaqfzbgaa8hi aiab zmigkab bcagiaa9fgadawgaaadbagaa8hdaaqabawccawfga gaey4jiktaa8hiaiaa hcacaqfubqeduudjxwakbyu51myvxgaicaa cqgesislcqgegaaicaqf0bgaa8xkaaqabagccawfgagaa8hkaiaa9d facawfoagaa0hdaiaa9tcacaqfzbgaa8xkaiaa bcacqgesislcqge gaaicaqfxbgaa8hkaiaa9rhacawfpagaa8xkaiaa9fgadaqhaawcba gaa0hdaaqaaiab6jhitiab6fdaxiab69caviab6jdayaaaaaa 83f2 mathop sup limits o leqq t leqq t mathop sup limits o s leqq a t wedge t t w t s w t a t 1 2

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

a differential geometric approach to nonlinear filtering the projection filter this paper presents a new and systematic method of approximating exact nonlinear filters with finite dimensional filters using the differential geometric approach to statistics the projection filter is defined rigorously in the case of exponential families a convenient exponential family is proposed which allows one to simplify the projection filter equation and to define an a posteriori measure of the local error of the projection filter approximation finally simulation results are discussed for the cubic sensor problem

exact convergence rates in strong approximation laws for large increments of partial sums consider partial sumss n of an i i d sequencex1x2 of centered random variables having a finite moment generating function in a neighborhood of zero the asymptotic behaviour of u n mathop max limits 0 leqq k leqq n b n s k b n s k is investigated where 1 b n n denotes an integer sequence such thatb n logn asn in particular ifb n o log p n asn for somep 1 the exact convergence rate ofu n b n n 1 0 1 is determined where n depends uponb n and the distribution ofx1 in addition a weak limit law foru n is derived finally it is shown how strong invariance takes over if mathop lim limits n to infty b n loglogn 2 log3n

improved erdos renyi and strong approximation laws for increments of partial sums let x 1 x 2 cdots be an i i d sequence with ex 1 0 ex 2 1 1 ee tx 1 infty t t 0 and partial sums s n x 1 cdots x n starting from some analogous results for the wiener process this paper studies the almost sure limiting behaviour of max 0 leq n leq n a n a 1 2 n s n a n s n as n rightarrow infty under various conditions on the integer sequence a n improvements of the erdos renyi law of large numbers for partial sums are obtained as well as strong invariance principle type versions via the komlos major tusnady approximation an appearing gap between these two results is also going to be closed

feedback control of quantum state reduction feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement we formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter we explore the use of stochastic lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability

conditional densities for continuous time nonlinear hybrid systems with applications to fault detection continuous time nonlinear stochastic differential state and measurement equations all of which have coefficients capable of abrupt changes at a random time are considered finite state jump markov chains are used to model the changes conditional probability densities which are essential in obtaining filtered estimates for these hybrid systems are then derived they are governed by a coupled system of stochastic partial differential equations when the q matrix of the markov chain is either lower or upper diagonal it is shown that the system of conditional density equations is finite dimensional computable these findings are then applied to a fault detection problem to compute state estimates that include the failure time

an improved quantum projection filter abstract this work extends the previous quantum projection filtering scheme in gao et al 2019 by adding an optimality analysis result a reformulation of the quantum projection filter is derived by minimizing the truncated stratonovich stochastic taylor expansion of the difference between the true quantum trajectory and its approximation on a lower dimensional submanifold through quantum information geometric techniques simulation results for a qubit example demonstrate better approximation performance for the new quantum projection filter

dynamics of the minority game for patients abstract we analyze the minority game for patients and the results known from the minority game are applied to the patient problem consulted at the department of pediatric cardiology we find numerically the standard deviation and the global efficiency which is discussed similar to the el farol bar problem after the score equation and the scaled utility are introduced the dynamical behavior of our model is discussed for particular strategies our results presented will be compared with recent numerical calculations

general large deviations of longest gaps in homogeneous poisson processes abstract let n t t 0 be a homogeneous poisson process with unit intensity the longest gap l t of this process is the maximal time subinterval between epochs of arrival times up to time t and it finds applications in the theory of reliability it has been known since the 1980s that the longest gap obeys the laws of large numbers l t ln t 1 as t almost surely however large deviation probabilities of the longest gap have not been well developed yet and in this paper we derive such large deviations with a very general speed the main tool is to establish a global estimation for the distribution function of the longest gap which is of independent interest

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

coherent observers for linear quantum stochastic systems the theory of observers is a basic part of classical linear system theory the purpose of this paper is to develop a theory of coherent observers for linear quantum systems we provide a class of coherent quantum observers which track the observables of a linear quantum stochastic system in the sense of mean values independent of any additional quantum noise in the observer we prove that there always exists such a coherent quantum observer described by quantum stochastic differential equations in the heisenberg picture and show how it can be designed to be consistent with the laws of quantum mechanics we also find a lower bound for the mean squared estimation error due to the uncertainty principle in addition we explore the quantum correlations between a linear quantum plant and the corresponding coherent observer it is shown that considering a joint plant observer gaussian quantum system entanglement can be generated under the condition that appropriate coefficients of the coherent quantum observer are chosen and this issue is illustrated in an example these results pave the way towards observer based quantum control

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

an exponential quantum projection filter for open quantum systems an approximate exponential quantum projection filtering scheme is developed for a class of open quantum systems described by hudson parthasarathy quantum stochastic differential equations aiming to reduce the computational burden associated with online calculation of the quantum filter by using a differential geometric approach the quantum trajectory is constrained in a finite dimensional differentiable manifold consisting of an unnormalized exponential family of quantum density operators and an exponential quantum projection filter is then formulated as a number of stochastic differential equations satisfied by the finite dimensional coordinate system of this manifold a convenient design of the differentiable manifold is also presented through reduction of the local approximation errors which yields a simplification of the quantum projection filter equations it is shown that the computational cost can be significantly reduced by using the quantum projection filter instead of the quantum filter it is also shown that when the quantum projection filtering approach is applied to a class of open quantum systems that asymptotically converge to a pure state the input to state stability of the corresponding exponential quantum projection filter can be established simulation results from an atomic ensemble system example are provided to illustrate the performance of the projection filtering scheme it is expected that the proposed approach can be used in developing more efficient quantum control methods

h infty control of linear quantum stochastic systems the purpose of this paper is to formulate and solve a h infin controller synthesis problem for a class of noncommutative linear stochastic systems which includes many examples of interest in quantum technology the paper includes results on the class of such systems for which the quantum commutation relations are preserved such a requirement must be satisfied in a physical quantum system a quantum version of standard classical dissipativity results are presented and from this a quantum version of the strict bounded real lemma is derived this enables a quantum version of the two riccati solution to the h infin control problem to be presented this result leads to controllers which may be realized using purely quantum purely classical or a mixture of quantum and classical elements this issue of physical realizability of the controller is examined in detail and necessary and sufficient conditions are given our results are constructive in the sense that we provide explicit formulas for the hamiltonian function and coupling operator corresponding to the controller

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

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm

statistical mechanics of systems with heterogeneous agents minority games we study analytically a simple game theoretical model of heterogeneous interacting agents we show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz such a stationary state differs from the nash equilibrium where each agent maximizes her own utility the latter turns out to be characterized by a replica symmetry broken structure numerical results fully agree with our analytical findings

an introduction to quantum filtering this paper provides an introduction to quantum filtering theory an introduction to quantum probability theory is given focusing on the spectral theorem and the conditional expectation as a least squares estimate and culminating in the construction of wiener and poisson processes on the fock space we describe the quantum ito calculus and its use in the modeling of physical systems we use both reference probability and innovations methods to obtain quantum filtering equations for system probe models from quantum optics

convolution roots and embeddings of probability measures on lie groups we show that for a large class of connected lie groups g viz from classc described below given a probability measure on g and a natural number n for any sequence i of nth convolution roots of there exists a sequence zi of elements of g centralising the support of and such that zi izi 1 is relatively compact thus the set of roots is relatively compact modulo the conjugation action of the centraliser of supp we also analyse the dependence of the sequence zi on n the results yield a simpler and more transparent proof of the embedding theorem for infinitely divisible probability measures on the lie groups as above proved in s g dani m mccrudden embeddability of infinitely divisible distributions on linear lie groups invent math 110 1992 237 261

feedback stabilization of discrete time quantum systems subject to non demolition measurements with imperfections and delays we consider a controlled quantum system whose finite dimensional state is governed by a discrete time nonlinear markov process in open loop the measurements are assumed to be quantum non demolition qnd the eigenstates of the measured observable are thus the open loop stationary states they are used to construct a closed loop supermartingale playing the role of a strict control lyapunov function the parameters of this supermartingale are calculated by inverting a metzler matrix that characterizes the impact of the control input on the kraus operators defining the markov process the resulting state feedback scheme taking into account a known constant delay provides the almost sure convergence to the target state this convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix closed loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box a cavity quantum electrodynamics system

direct and indirect couplings in coherent feedback control of linear quantum systems the purpose of this paper is to study and design direct and indirect couplings for use in coherent feedback control of a class of linear quantum stochastic systems a general physical model for a nominal linear quantum system coupled directly and indirectly to external systems is presented fundamental properties of stability dissipation passivity and gain for this class of linear quantum models are presented and characterized using complex lyapunov equations and linear matrix inequalities lmis coherent h and lqg synthesis methods are extended to accommodate direct couplings using multistep optimization examples are given to illustrate the results

erdős renyi laws for gibbs measures can one detect a phase transition from a single large sample of a gibbs measure what information does one get on the other gibbs distributions with the same potential we approach these questions via erd s renyi laws in particular we prove almost sure limit theorems for sets of empirical distributions of sub samples of the given one for suitable sub samples size this set converges to the set of stationary gibbs measures moreover we formulate erd s renyi laws for general families of random variables with suitable large deviation principles

quasi free quantum stochastic integrals for the car and ccr abstract a theory of quantum martingales and quantum stochastic integrals in quasi free representations of the car and ccr is presented for the car the results generalize some of those developed in barnett streater and wilde j funct anal 48 1982 172 212 j london math soc 27 1983 373 384 and for the ccr the results contain the standard ito theory of stochastic integration with respect to brownian motion as a special case