quantum macrostates equivalence of ensembles and an h theorem before the thermodynamic limit macroscopic averages need not commute for a quantum system as a consequence aspects of macroscopic fluctuations or of constrained equilibrium require a careful analysis when dealing with several observables we propose an implementation of ideas that go back to john von neumann s writing about the macroscopic measurement we apply our scheme to the relation between macroscopic autonomy and an h theorem and to the problem of equivalence of ensembles in particular we show how the latter is related to the asymptotic equipartition theorem the main point of departure is an expression of a law of large numbers for a sequence of states that start to concentrate as the size of the system gets larger on the macroscopic values for the different macroscopic observables deviations from that law are governed by the entropy

an ergodic theorem for the quantum relative entropy we prove the ergodic version of the quantum stein s lemma which was conjectured by hiai and petz the result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability and contains as a special case the quantum version of the shannon mcmillan theorem for ergodic states a version of the quantum relative asymptotic equipartition property aep is given

large deviations for quantum spin systems we consider high temperature kms states for quantum spin systems on a lattice we prove a large deviation principle for the distribution of empirical averages bar x lambda frac 1 lambda sigma i in lambda x i where the x i s are copies of a self adjoint element x level one large deviations from the analyticity of the generating function we obtain the central limit theorem we generalize to a level two large deviation principle for the distribution of frac 1 lambda sigma i in lambda delta x i

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

a computer aided test for the absence of limit cycles in fixed point digital filters the problem of determining the existence of limit cycles in direct form digital filters with quantization nonlinearities is considered a method based on an exhaustive search is introduced it is applicable to all direct form digital filters regardless of the type of quantization arithmetic the number of quantizers and the order of the filter the equivalence between the concept of global asymptotic stability and the exclusion of limit cycles is proved for linearly stable finite wordlength digital filters the robustness of this test with respect to coefficient perturbation in the digital filter is presented the computational aspects of the algorithm are also discussed the algorithm is shown to be capable of always determining the global asymptotic stability of the tested realization

equivalence classes and local asymptotic normality in system identification for quantum markov chains we consider the problem of identifying and estimating dynamical parameters of an ergodic quantum markov chain when only the stationary output is accessible for measurements the starting point of the analysis is the fact that the knowledge of the output state completely fixes the dynamics up to an equivalence class of coordinate transformation consisting of a multiplication by a phase and a unitary conjugation of the kraus operators

homogeneous open quantum random walks on a lattice we study open quantum random walks oqrws for which the underlying graph is a lattice and the generators of the walk are homogeneous in space using the results recently obtained in carbone and pautrat ann henri poincare 2015 we study the quantum trajectory associated with the oqrw which is described by a position process and a state process we obtain a central limit theorem and a large deviation principle for the position process we study in detail the case of homogeneous oqrws on the lattice mathbb z d with internal space mathfrak h mathbb c 2

bibo stability of multidimensional md shift invariant discrete systems bibo stability of 1 d and multidimensional md shift varying discrete systems is analyzed the test for one particular shift varying md system involves one md linear shift invariant stability test only the stability conditions derived also provides insight into robustness and margin of stability of the shift varying system two classes of systems are investigated 1 shift varying discrete direct realization systems and 2 shift varying discrete state space realization systems the results extend an implication of the perron frobenius theorem from 1 d shift invariant systems to shift invariant and or multidimensional systems

large deviations and chernoff bound for certain correlated states on a spin chain in this paper we extend the results of lenci and rey bellet j stat phys 119 715 2005 on the large deviation upper bound of the distribution measures of local hamiltonians with respect to a gibbs state in the setting of translation invariant finite range interactions we show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by gibbs states of the above kind as well as finitely correlated states as an application of the methods the chernoff bound for correlated states with factorization property is studied in the specific case of the distributions of the ergodic averages of a one site observable with respect to an ergodic finitely correlated state the spectral theory of positive maps is applied to prove the full large deviation principle

a stability analysis of two dimensional nonlinear digital state space filters two different methods for a stability analysis of 2 d nonlinear discrete state space systems under zero input conditions are provided the first method reduces the task of testing a 2 d discrete nonlinear or shift varying system to a single 2 d linear stability test of a system matrix with nonnegative system matrix entries the second method is based on a number of norm tests for products of extreme matrices and can be considered the 2 d counterpart of the method for 1 d systems described by k t erickson and a n michel 1985 both of the introduced methods are based on the sector description of the nonlinearity and can be used to analyze digital filter stability under finite word length effects

a quantum version of sanov s theorem we present a quantum version of sanov s theorem focussing on a hypothesis testing aspect of the theorem there exists a sequence of typical subspaces for a given set of stationary quantum product states asymptotically separating them from another fixed stationary product state analogously to the classical case the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set while in the classical case the separating subsets can be chosen universally in the sense that they depend only on the chosen set of i i d processes in the quantum case the choice of the separating subspaces depends additionally on the reference state

fisher informations and local asymptotic normality for continuous time quantum markov processes we consider the problem of estimating an arbitrary dynamical parameter of an open quantum system in the input output formalism for irreducible markov processes we show that in the limit of large times the system output state can be approximated by a quantum gaussian state whose mean is proportional to the unknown parameter this approximation holds locally in a neighbourhood of size in the parameter space and provides an explicit expression of the asymptotic quantum fisher information in terms of the markov generator furthermore we show that additive statistics of the counting and homodyne measurements also satisfy local asymptotic normality and we compute the corresponding classical fisher informations the general theory is illustrated with the examples of a two level system and the atom maser our results contribute towards a better understanding of the statistical and probabilistic properties of the output process with relevance for quantum control engineering and the theory of non equilibrium quantum open systems

optimal sequence of quantum measurements in the sense of stein s lemma in quantum hypothesis testing we derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing we discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0 as an asymptotically optimal measurement we propose a projection measurement characterized by the irreducible representation theory of the special linear group sl especially in the spin 1 2 system it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction as a by product we obtain another proof of quantum stein s lemma in addition an asymptotically optimal measurement is constructed in the quantum gaussian case and it is physically meaningful

strong converse and stein s lemma in quantum hypothesis testing the hypothesis testing problem for two quantum states is treated we show a new inequality between the errors of the first kind and the second kind which complements the result of hiai and petz 1991 to establish the quantum version of stein s lemma the inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy which yields the strong converse in quantum hypothesis testing finally we discuss the relation between the bound and the power exponent derived by han and kobayashi 1989 in classical hypothesis testing

two s complement quantization in two dimensional state space digital filters two dimensional 2 d state space digital filters are investigated for stability with and without quantization a sufficient condition is obtained for the stability of the linear fornasini marchesini 1976 state space model the same model is then studied for stability when implemented using two s complement truncation quantization and a sufficient condition for asymptotic stability of the nonlinear filter is obtained in the process a theorem is proved which gives a sufficient condition for the stability of a one dimensional 1 d state space digital filter under the same type of quantization

asymptotics of quantum relative entropy from a representation theoretical viewpoint in this paper it is proved that the quantum relative entropy d can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive operator valued measures povms the sequence of povms depends on but is independent of the choice of

null controllability of systems with control constraints and state saturation abstract sufficient conditions for the null controllability of discrete time systems with control constraints and state saturation are presented this is accomplished in two stages first we establish a result for the asymptotic stability of such systems under zero input conditions next we utilize this stability result to obtain a condition for the null controllability of such systems we include specific examples to demonstrate the applicability of the present results

entropy densities for algebraic states abstract algebraic or finitely correlated states are translation invariant states on an infinite tensor product c algebra whose construction is rather general including quantum markov chains for a strongly mixing algebraic state we obtain the relation between the mean entropy and another entropy density which means the macroscopic uniformity under the relevant state in the statistical mechanical sense for this purpose an outstanding property of approximately product type is shown for strongly mixing algebraic states we also obtain similar relations of the mean relative entropy to other entropy densities for two translation invariant states when the reference one is a strongly mixing algebraic state

typical support and sanov large deviations of correlated states discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support the exponential growth rate of which is given by the maximal ergodic entropy in the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy according to sanov s theorem we give an extension to the correlated case referring to the newly introduced class of hp states

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

suppression of limit cycles in the first order two dimensional direct form digital filter with a controlled rounding arithmetic the first order two dimensional direct form digital filter with magnitude truncation is known to be free from limit cycles for a limited range of allowed filter coefficients in this correspondence the quantization technique of controlled rounding is used to extend the region of filter coefficients for which suppression of limit cycles can be proved with controlled rounding a signal is quantized in the direction of a control signal which is formed by an integer combination of preceding signal values the absence of limit cycles is proved with a properly chosen lyapunov function

on the bose trautman condition for stability of two dimensional linear systems in this correspondence we give a counterexample that shows that the bose trautman condition for the stability of 2 d linear models is incorrect the correction and equivalent conditions are given together with simple proofs from non negative matrix theory

on the limit point classification of a class of non self adjoint ordinary differential operators the limit point classification of a class of non symmetric ordinary differential expressions on half line is considered which is related to the well posedness of certain boundary value problems

an information spectrum approach to classical and quantum hypothesis testing for simple hypotheses the information spectrum analysis made by han for classical hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and quantum hypothesis testing the results are also applied to fixed length source coding when loosening the normalizing condition for probability distributions and for quantum states we establish general formulas for several quantities relating to the asymptotic optimality of tests codes in terms of classical and quantum information spectra

exponential error rates in multiple state discrimination on a quantum spin chain we consider decision problems on finite sets of hypotheses represented by pairwise different shift invariant states on a quantum spin chain the decision in favor of one of the hypotheses is based on outcomes of generalized measurements performed on local states on blocks of finite size we assume the existence of the mean quantum chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum chernoff distance we establish that this minimum specifies an asymptotic bound on the exponential rate of decay of the averaged probability of rejecting the true state in increasing block size if the mean quantum chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem this assumption is in particular fulfilled by shift invariant product states independent and identically distributed states further we provide a constructive proof for the existence of a sequence of quan

a quantum version of wielandt s inequality in this paper wielandt s inequality for classical channels is extended to quantum channels that is an upper bound to the number of times a channel must be applied so that it maps any density operator to one with full rank is found using this bound dichotomy theorems for the zero error capacity of quantum channels and for the matrix product state mps dimension of ground states of frustration free hamiltonians are derived the obtained inequalities also imply new bounds on the required interaction range of hamiltonians with unique mps ground state

the shannon mcmillan theorem for ergodic quantum lattice systems we formulate and prove a quantum shannon mcmillan theorem the theorem demonstrates the significance of the von neumann entropy for translation invariant ergodic quantum spin systems on lattices the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes the one dimensional case covers quantum information sources and is basic for coding theorems

new criteria for asymptotic stability of one and multidimensional state space digital filters in fixed point arithmetic this paper addresses the problem of global asymptotic stability of one dimensional 1 d and multidimensional m d digital filters with any combination of overflow and quantization nonlinearities the stability analysis is carried out using 1 d and m d state space representations the approach introduced allows one to determine the stability behavior of single input single output systems with overflow and quantization nonlinearities the new criteria based on previous stability results of digital filters with quantization schemes are applicable to all arithmetic schemes for the first time results concerning general state variable representations of 1 d and m d digital filters with the naturally occurring combination of two s complement truncation quantization and overflow are reported furthermore significantly improved stability regions are obtained for digital filters with roundoff nonlinearities

asymptotic and non asymptotic analysis for a hidden markovian process with a quantum hidden system we focus on a data sequence produced by repetitive quantum measurement on an internal hidden quantum system and call it a hidden markovian process using a quantum version of the perron frobenius theorem we derive novel upper and lower bounds for the cumulant generating function of the sample mean of the data using these bounds we derive the central limit theorem and large and moderate deviations for the tail probability then we give the asymptotic variance by using the second derivative of the cumulant generating function we also derive another expression for the asymptotic variance by considering the quantum version of the fundamental matrix further we explain how to extend our results to a general probabilistic system

large deviations in quantum spin chains we show the full large deviation principle for kms states and c finitely correlated states on a quantum spin chain we cover general local observables our main tool is ruelle s transfer operator method

large deviations for ideal quantum systems we consider a general d dimensional quantum system of non interacting particles in a very large formally infinite container we prove that in equilibrium the fluctuations in the density of particles in a subdomain of the container are described by a large deviation function related to the pressure of the system that is untypical densities occur with a probability exponentially small in the volume of with the coefficient in the exponent given by the appropriate thermodynamic potential furthermore small fluctuations satisfy the central limit theorem

sanov and central limit theorems for output statistics of quantum markov chains in this paper we consider the statistics of repeated measurements on the output of a quantum markov chain we establish a large deviations result analogous to sanov s theorem for the multi site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process our result relies on the construction of an extended quantum transition operator which keeps track of previous outcomes in terms of which we compute moment generating functions and whose spectral radius is related to the large deviations rate function as a corollary to this we obtain a central limit theorem for the empirical measure such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions which are not captured by lower level statistics such as the sample mean as a step in this direction we give an example of a finite system whose level 1 empirical mean rate function is independent of a model parameter while the level 2 empirical measure rate is not

stability analysis of multidimensional m d direct realization digital filters under the influence of nonlinearities a stability test of m d direct realization digital filters with nonlinearities such as quantization overflow saturation etc is presented the method presented reduces a given nonlinear m d system to a nonlinear 1 d system such that the stability of the latter is sufficient for the stability of the former to check stability of the nonlinear 1 d system some new results on 1 d recursive systems with positive coefficients are applied the case of multiple nonlinearities in multidimensions is also addressed examples are given illustrating the application of the method

stability and overflow oscillations in 2 d state space digital filters an important theorem relating to the lyapunov stability of two dimensional discrete systems is proven using this theorem it is shown that for any 2 d digital filter satisfying shanks criterion there exists a realization that cannot support overflow oscillations in the process of proving the theorem some interesting results on the multi dimensional bilinear transformation are developed one of these results yields a simple test that can be used to check the stability of a 2 d discrete transfer function that has been obtained from the bilinear transform of a 2 d continuous transfer function with a 2 d hurwitzian denominator polynomial a technique is given for determining whether a normal realization exists for a given 2 d discrete system also a theorem is presented that allows the determination of the norm of the minimum norm realization of a given transfer function a noniterative technique for obtaining a low norm realization and an iterative technique for obtaining a minimum norm realization are developed

error exponents in hypothesis testing for correlated states on a spin chain we study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum chernoff and hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain our results apply to states satisfying a certain factorization property typical examples are the global gibbs states of translation invariant finite range interactions as well as certain finitely correlated states

on deformations approximations and nonlinear filtering given a nonlinear filtering problem there is associated to it a lie algebra l of differential operators which is the lie algebra generated by the two operators occurring in the zakai equation for the unnormalized conditional density of the problem the representability of l or quotients of l by means of vectorfields on a finite dimensional manifold is strongly related to the existence of exact finite dimensional recursive filters however in many cases including the cubic sensor problem and the problem d x d w d z x x 3 d t d 0 the algebra l is isomorphic to the weyl algebra w l x d dx which admits no nonzero homomorphisms into any lie algebra of vectorfields on a finite dimensional manifold on the other hand the lie algebra l mod n is finite dimensional for all n which opens up the possibility of the existence of a sequence of converging approximate filters