indirect acquisition of information in quantum mechanics long sequences of successive direct projective measurements or observations of just a few uninteresting physical quantities pertaining to a quantum system such as clicks of some detectors may reveal indirect but precise and unambiguous information on the values of some very interesting observables of the system in this paper the mathematics underlying this claim is developed i e we attempt to contribute to a mathematical theory of indirect and in particular non demolition observations and measurements in quantum mechanics our attempt leads us to make some novel uses of classical notions and results of probability theory such as the algebra of functions measurable at infinity the central limit theorem results concerning relative entropy and its role in the theory of large deviations etc

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

random positive operator valued measures we introduce several notions of random positive operator valued measures povms and we prove that some of them are equivalent we then study statistical properties of the effect operators for the canonical examples starting from the limiting eigenvalue distribution we derive the large system limit for several quantities of interest in quantum information theory such as the sharpness the noise content and the probability range finally we study different compatibility criteria and we compare them for generic povms

general twisting of algebras abstract we introduce the concept of pseudotwistor with particular cases called twistor and braided twistor for an algebra a u in a monoidal category as a morphism t a a a a satisfying a list of axioms ensuring that a t u is also an algebra in the category this concept provides a unifying framework for various deformed or twisted algebras from the literature such as twisted tensor products of algebras twisted bialgebras and algebras endowed with fedosov products pseudotwistors appear also in other topics from the literature e g durdevich s braided quantum groups and ribbon algebras we also focus on the effect of twistors on the universal first order differential calculus as well as on lifting twistors to braided twistors on the algebra of universal differential forms

iterated twisted tensor products of nonlocal vertex algebras abstract this is the second paper in a series devoted to studies of twisted tensor products of nonlocal vertex algebras in this paper we introduce and study iterated twisted tensor products of nonlocal vertex algebras among the main results we find conditions for constructing an iterated twisted tensor product of three factors and prove that those conditions are enough for building an iterated twisted tensor product of any number of factors and we also establish a universal property and give a characterization of an iterated twisted tensor product furthermore we give an example of iterated twisted tensor product nonlocal vertex algebra

contractivity of positive and trace preserving maps under lp norms we provide a complete picture of contractivity of trace preserving positive maps with respect to p norms we show that for p 1 contractivity holds in general if and only if the map is unital when the domain is restricted to the traceless subspace of hermitian matrices then contractivity is shown to hold in the case of qubits for arbitrary p 1 and in the case of qutrits if and only if p 1 in all noncontractive cases best possible bounds on the p norms are derived

common eigenvectors of two matrices abstract a computable criterion is given for two square matrices to possess a common eigenvector as well as a criterion for one matrix to have an eigenvector lying in a given subspace some applications are discussed

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

iterated stochastic measurements we describe a measurement device principle based on discrete iterations of bayesian updating of system state probability distributions although purely classical by nature these measurements are accompanied with a progressive collapse of the system state probability distribution during each complete system measurement this measurement scheme finds applications in analysing repeated non demolition indirect quantum measurements we also analyse the continuous time limit of these processes either in the brownian diffusive limit or in the poissonian jumpy limit in the quantum mechanical framework this continuous time limit leads to belavkin s equations which describe quantum systems under continuous measurements this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

quantum stochastic processes as models for state vector reduction an elementary introduction of quantum state valued markovian stochastic processes qsp for n state quantum systems is given it is pointed out that a so called master constraint must be fulfilled for a given master equation a continuous and as a new alternative possibility a discontinuous qsp are derived both are discussed as possible models for state reduction during measurement

the nodal line of the second eigenfunction of the robin laplacian in r2 can be closed abstract we construct a multiply connected domain in r 2 for which the second eigenfunction of the laplacian with robin boundary conditions has an interior nodal line in the process we adapt a bound of donnelly fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains which does not depend directly on any boundary behaviour this also gives a new proof of the nodal line property of the example in the dirichlet case

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

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

large time behavior and convergence rate for quantum filters under standard non demolition conditions a quantum system mathcal s undergoing continuous time measurement is usually described by a jump diffusion stochastic differential equation such an equation is called a quantum filtering equation or quantum stochastic master equation and its solution is called a quantum filter or quantum trajectory this solution describes the evolution of the state of mathcal s in the context of quantum non demolition measurement we investigate the large time behavior of this solution it is rigorously shown that for large time this solution behaves as if a direct von neumann measurement has been performed at time 0 in particular the solution converges to a random pure state which can be directly linked to the wave packet reduction postulate using the theory of girsanov transformation we obtain the explicit rate of convergence towards this random state the problem of state estimation used in experiment is also investigated

a continuous counting observation and posterior quantum dynamics a stochastic model for a continuous photon counting measurement is proposed an equation for the generating functional of the statistics of counts is found a stochastic linear equation for the unnormalised posterior state vector of atoms continuously collapsing without mixing is derived by using quantum stochastic calculus methods

eigenvalue inequalities for the laplacian with mixed boundary conditions abstract inequalities for the eigenvalues of the negative laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established the eigenvalues subject to a dirichlet boundary condition on a part of the boundary and a neumann boundary condition on the remainder of the boundary are estimated in terms of either dirichlet or neumann eigenvalues the results complement several classical inequalities between dirichlet and neumann eigenvalues due to polya payne levine and weinberger friedlander and others

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

continuum limits of homogeneous binary trees and the thompson group tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching conformal field theory cft predictions in the continuum limit it is natural to seek a more complete correspondence additionally incorporating dynamics on the cft side this is determined by a representation of the diffeomorphism group of the circle in a remarkable series of papers jones outlined a research program where the thompson group t takes the role of the latter in the discrete setting and representations of t are constructed from certain elements of a subfactor planar algebra he also showed that for a particular example of such a construction this approach only yields in the continuum limit a representation which is highly discontinuous and hence unphysical here we show that the same issue arises generically when considering tree tensor networks the set of coarse graining maps yielding discontinuous representations has full measure in the set of all isometries this extends jones s no go example to typical elements of the so called tensor planar algebra we also identify an easily verified necessary condition for a continuous limit to exist this singles out a particular class of tree tensor networks our considerations apply to recent approaches for introducing dynamics in holographic codes

the peripheral point spectrum of schwarz operators on c algebras the spectral theory of positive operators has its origin in the classical work of perron and frobenius who considered matrices with positive entries on finitedimensional vector spaces and obtained results on the cyclic structure of the set of eigenvalues of such operators frobenius main result rests on the concept of irreducibility which is usually defined in matrix terms but it is easy to see that this concept involves the ideals of the commutative c algebra c n in this paper we extend these results to the case of positive operators on arbitrary c algebras the extensions to commutative c algebras has been studied among many others by schaefer 10 we shall now briefly discuss the content of this paper in sect 1 and 2 we introduce the basic definitions e g irreducibility and present the classical results of krein rutman on the peripheral point spectrum of the adjoint of a positive operator in the context of c algebras in sect 3 we consider a c algebra 96 and a linear operator t on 9 1 satisfying t x x t x t x for all x e n and t i l we show that the results of frobenius can be extended to irreducible operators of that type we also study the action of t on the algebra generated by the eigenvectors pertaining to the peripheral eigenvalues of t

asymptotics of repeated interaction quantum systems abstract a quantum system s interacts in a successive way with elements e of a chain of identical independent quantum subsystems each interaction lasts for a duration and is governed by a fixed coupling between s and e we show that the system initially in any state close to a reference state approaches a repeated interaction asymptotic state in the limit of large times this state is periodic in time and does not depend on the initial state if the reference state is chosen so that s and e are individually in equilibrium at positive temperatures then the repeated interaction asymptotic state satisfies an average second law of thermodynamics

random repeated quantum interactions and random invariant states we consider a generalized model of repeated quantum interactions where a system mathcal h is interacting in a random way with a sequence of independent quantum systems mathcal k n n geq 1 two types of randomness are studied in detail one is provided by considering haar distributed unitaries to describe each interaction between mathcal h and mathcal k n the other involves random quantum states describing each copy mathcal k n in the limit of a large number of interactions we present convergence results for the asymptotic state of mathcal h this is achieved by studying spectral properties of random quantum channels which guarantee the existence of unique invariant states finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble

on some classes of bipartite unitary operators we investigate unitary operators acting on a tensor product space with the property that the quantum channels they generate via the stinespring dilation theorem are of a particular type independently of the state of the ancilla system in the stinespring relation the types of quantum channels we consider are those of interest in quantum information theory unitary conjugations constant channels unital channels mixed unitary channels ppt channels and en tanglement breaking channels for some of the classes of bipartite unitary operators corresponding to the above types of channels we provide explicit characterizations necessary and or sufficient conditions for membership and we compute the dimension of the corresponding algebraic variety inclusions between these classes are considered and we show that for small dimensions many of these sets are identical

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

hardy inequalities for robin laplacians abstract in this paper we establish a hardy inequality for laplace operators with robin boundary conditions for convex domains in particular we show explicitly how the corresponding hardy weight depends on the coefficient of the robin boundary conditions we also study several extensions to non convex and unbounded domains

strict inequality of robin eigenvalues for elliptic differential operators on lipschitz domains abstract on a bounded lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to robin boundary conditions where the coefficients in the boundary conditions are functions we prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty open set

twisted tensor products of nonlocal vertex algebras abstract in this paper we introduce and study a twisted tensor product construction of nonlocal vertex algebras among the main results we establish a universal property and give a characterization of a twisted tensor product furthermore we give a construction of modules for a twisted tensor product we also show that smash products studied by one of us before can be realized as twisted tensor products

common invariant subspaces of two matrices abstract a finite rational procedure of the shemesh type is proposed to check whether given complex n n matrices a and b have a common invariant subspace of dimension k 2 k n the only restriction is that at least one of a and b must have distinct eigenvalues this fact can also be verified by a finite rational computation

stabilization of a delayed quantum system the photon box case study a feedback scheme stabilizing an arbitrary photon number state in a microwave cavity is analyzed the quantum non demolition measurement of the cavity state allows in open loop a non deterministic preparation of photon number states by the mean of a controlled classical field injection this preparation process is made deterministic the system evolves through a discrete time markov process and the feedback law relies on lyapunov techniques this feedback design compensates an unavoidable pure delay by a stochastic version of a kalman type predictor after illustrating the efficiency of the proposed feedback law through simulations the global closed loop convergence is proved it relies on tools from stochastic stability analysis a brief study of the lyapunov exponents of the linearized system around the target state gives a strong indication of the robustness of the method

a remark on schatten von neumann properties of resolvent differences of generalized robin laplacians on bounded domains in this note we investigate the asymptotic behavior of the s numbers of the resolvent difference of two generalized self adjoint maximal dissipative or maximal accumulative robin laplacians on a bounded domain with smooth boundary for this we apply the recently introduced abstract notion of quasi boundary triples and weyl functions from extension theory of symmetric operators together with krein type resolvent formulae and well known eigenvalue asymptotics of the laplace beltrami operator on it is shown that the resolvent difference of two generalized robin laplacians belongs to the schatten von neumann class of any order p for which

robin boundary value problems on arbitrary domains we develop a theory of generalised solutions for elliptic boundary value problems subject to robin boundary conditions on arbitrary domains which resembles in many ways that of the dirichlet problem in particular we establish lp lq estimates which turn out to be the best possible in that framework we also discuss consequences to the spectrum of robin boundary value problems finally we apply the theory to parabolic equations

application of shemesh theorem to quantum channels completely positive maps are useful in modeling the discrete evolution of quantum systems spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel we discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its kraus operators a a1 ak by applying the shemesh and amitsur levitzki theorems to analyse the structure of the algebra a a1 ak one can predict the asymptotic dynamics for a class of operations completely positive maps are useful in modeling the discrete evolution of quantum systems spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel we discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its kraus operators a a1 ak by applying the shemesh and amitsur levitzki theorems to analyse the structure of the algebra a a1 ak one can predict the asymptotic dynamics for a class of operations

boundary value problems for elliptic partial differential operators on bounded domains for a symmetric operator or relation a with infinite deficiency indices in a hilbert space we develop an abstract framework for the description of symmetric and self adjoint extensions a of a as restrictions of an operator or relation t which is a core of the adjoint a this concept is applied to second order elliptic partial differential operators on smooth bounded domains and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

twistors of nonlocal vertex algebras abstract in this paper we introduce and study the concept of twistor for a nonlocal vertex algebra this concept provides a unified framework for various constructions of nonlocal vertex algebras such as r matrices twisted tensor products iterated twisted tensor products and l r twisted tensor products of nonlocal vertex algebras among the main results we find the relations among theses constructions and we also give one concrete example of a twistor furthermore we study some properties of twistors

nonlocal robin laplacians and some remarks on a paper by filonov on eigenvalue inequalities abstract the aim of this paper is twofold first we characterize an essentially optimal class of boundary operators which give rise to self adjoint laplacians in l 2 d n x with nonlocal and local robin type boundary conditions on bounded lipschitz domains r n n n n 2 second we extend friedlander s inequalities between neumann and dirichlet laplacian eigenvalues to those between nonlocal robin and dirichlet laplacian eigenvalues associated with bounded lipschitz domains following an approach introduced by filonov for this type of problems

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

random repeated interaction quantum systems we consider a quantum system mathcal s interacting sequentially with independent systems mathcal e m m 1 2 before interacting each mathcal e m is in a possibly random state and each interaction is characterized by an interaction time and an interaction operator both possibly random we prove that any initial state converges to an asymptotic state almost surely in the ergodic mean provided the couplings satisfy a mild effectiveness condition we analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics

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

on the dimension of subspaces with bounded schmidt rank we consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states this is motivated in part by results of hayden et al e print arxiv quant ph 0407049 commun math phys 265 95 2006 which show that in large d d dimensional systems there exist random subspaces of dimension almost d2 all of whose states have entropy of entanglement at least logd o 1 it is also a generalization of results on the dimension of completely entangled subspaces which have connections with the construction of unextendible product bases here we take as entanglement measure the schmidt rank and determine for every pair of local dimensions da and db and every r the largest dimension of a subspace consisting only of entangled states of schmidt rank r or larger this exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in hayden et al we also determine the converse the l

a symplectic perspective on constrained eigenvalue problems abstract the maslov index is a powerful tool for computing spectra of selfadjoint elliptic boundary value problems this is done by counting intersections of a fixed lagrangian subspace which designates the boundary conditions with the set of cauchy data for the differential operator we apply this methodology to constrained eigenvalue problems in which the operator is restricted to a not necessarily invariant subspace the maslov index is defined and used to compute the morse index of the constrained operator we then prove a constrained morse index theorem which says that the morse index of the constrained problem equals the number of constrained conjugate points counted with multiplicity and give an application to the nonlinear schrodinger equation

diffusion approximation for a class of transport processes with physical reflection boundary conditions consider a stochastic process consisting of the pair of a position and a velocity in a piecewise el d dimensional domain in the interior of the domain the dynamics are assigned by a potential and by random changes of the velocity occurring at exponentially distributed times according to a probability distribution which may depend on the current position and velocity on the boundary the process reflects physically the angle of reflection equals the angle of incidence first it is shown that the process is well defined for all times then when the coefficients depend on a diverging parameter n in particular such that the speed and the jump rate of the velocity go to xo with order rn and at least n respectively a diffusion approximation is sought the position process is represented as a solution of a skorohod reflection equation a skewing effect on the boundary results from the interaction between the dynamics and the reflection law so that the direction of reflection is in general oblique the assumption that the mean change of the velocity in the interior is linear in the current velocity up to order at least 1 2 in 1 n ensures that the cone of directions of reflection is independent of n the continuity properties of the skorohod oblique reflection problem enable one to show tightness of the position processes without having to estimate explicitly local times boundary layers and together with a suitable law of large numbers for the velocity allow one to identify the limit stochastic differential equation with oblique reflection the theory is illustrated by several applications in particular one to a mechanical model of brownian motion

generic local distinguishability and completely entangled subspaces a subspace of a multipartite hilbert space is completely entangled if it contains no product states such subspaces can be large with a known maximum size smax approaching the full dimension of the system d we show that almost all subspaces with dimension s smax are completely entangled and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n d smax this condition holds for almost all sets of states of all multipartite systems and reveals something surprising the criterion is identical for separable and nonseparable states entanglement makes no difference

diffusion approximation of stochastic master equations with jumps in the presence of quantum measurements with direct photon detection the evolution of open quantum systems is usually described by stochastic master equations with jumps heuristically diffusion models can be obtained from these equations as an approximation a condition for a general diffusion approximation for jump master equations is presented this approximation is rigorously proved by using techniques for markov processes which are based on the convergence of markov generators and martingale problems this result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

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

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

fidelity is a sub martingale for discrete time quantum filters fidelity is known to increase through any kraus map the fidelity between two density matrices is less than the fidelity between their images via a kraus map we prove here that in average fidelity is also increasing for discrete time quantum filters attached to an arbitrary kraus map fidelity between the density matrix of the underlying markov chain and the density matrix of the associated quantum filter is a sub martingale this result is not restricted to pure states it also holds true for mixed states

complex obtuse random walks and their continuous time limits we study a particular class of complex valued random variables and their associated random walks complex obtuse random variables they generalize to the complex case the real valued obtuse random variables introduced in equations de structure pour des martingales vectorielles seminaire de probabilites xxviii p 256278 lecture notes in math vol 1583 springer berlin 1994 in order to understand the structure of normal martingales in mathbb r n the extension to the complex case is motivated by quantum statistical mechanics in particular for characterizing those quantum baths acting as classical noises the extension of obtuse random variables to the complex case is far from obvious and makes use of very interesting algebraical structures we show that complex obtuse random variables are characterized by a 3 tensor which admits certain symmetries we show that these symmetries are the exact 3 tensor analogue of the normal character for 2 tensors i e matrices that is a necessary and sufficient condition for being diagonalizable in some orthonormal basis we discuss the passage to the continuous time limit for these random walks and show that they converge in distribution to normal martingales in mathbb c n we show that the 3 tensor associated to these normal martingales encodes their behavior in particular the diagonalization directions of the 3 tensor indicate the directions of the space where the martingales behaves like a diffusion and those where it behaves like a poisson process we finally prove the convergence in the continuous time limit of the corresponding multiplication operators on the canonical fock space with an explicit expression in terms of the associated 3 tensor again

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

counterexamples to the maximal p norm multiplicativity conjecture for all p 1 for all p 1 we demonstrate the existence of quantum channels with non multiplicative maximal output p norms equivalently for all p 1 the minimum output renyi entropy of order p of a quantum channel is not additive the violations found are large in all cases the minimum output renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors since p 1 corresponds to the von neumann entropy these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true it cannot be proved as a consequence of any channel independent guarantee of maximal p norm multiplicativity we also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p 2