quantum error correction on infinite dimensional hilbert spaces we present a generalization of quantum error correction to infinite dimensional hilbert spaces we find that under relatively mild conditions much of the structure known from systems in finite dimensional hilbert spaces carries straightforwardly over to infinite dimensions we also find that at least in principle there exist qualitatively new classes of quantum error correcting codes that have no finite dimensional counterparts we begin with a shift of focus from states to algebras of observables standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite dimensional von neumann factors of type i the new classes of codes that arise in infinite dimensions are shown to be characterized by von neumann algebras of types ii and iii for which we give in principle physical examples

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

further results on entropy and separability let be a bi stochastic quantum operation we obtain that s s for any quantum state if and only if there exists a unitary operator u such that u u that is is a unitary operation it is well known that a bipartite quantum state ab is separable if and only if it can be written as ab i i i i i i where i i 1 i and i are the normalized pure states of the subsystems and respectively which may not be orthogonal in general in this paper we also present some necessary and sufficient conditions for which the sets i are orthonormal vectors

a quantum computing primer for operator theorists this is an exposition of some of the aspects of quantum computation and quantum information that have connections with operator theory after a brief introduction we discuss quantum algorithms we outline basic properties of quantum channels or equivalently completely positive trace preserving maps the main theorems for quantum error detection and correction are presented and we conclude with a description of a particular passive method of quantum error correction

sequential quantum measurements a quantum effect is an operator a on a complex hilbert space h that satisfies 0 a i we denote the set of quantum effects by e h the set of self adjoint projection operators on h corresponds to sharp effects and is denoted by p h we define the sequential product of a b e h by a b a1 2ba1 2 the main purpose of this article is to study some of the algebraic properties of the sequential product many of our results show that algebraic conditions on a b imply that a and b commute for the usual operator product for example if a b satisfies certain distributive or associative laws then ab ba moreover if a b p h then ab ba and a b b or b a b if and only if ab ba b a natural definition of stochastic independence is introduced and briefly studied

structural characterization and condition for measurement statistics preservation of a unital quantum operation we investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation acting on finite dimensional quantum states by reducing this problem to the problem of simultaneous diagonalization of the kraus operators associated with we can completely characterize the kinds of quantum states that are fixed by our work has several applications it gives a simple proof of the structural characterization of a unital quantum operation that acts on finite dimensional quantum states a result not explicitly mentioned in earlier studies it also provides a necessary and sufficient condition for determining what kind of measurement statistics is preserved by a unital quantum operation finally our result clarifies and extends the work of st rmer by giving a proof of a reduction theorem on the unassisted and entanglement assisted classical capacities coherent information and minimal output renyi entropy of a unital channel acting on a finite dimensional quantum state

noncommutative poisson boundaries of unital quantum operations in this paper poisson boundaries of unital quantum operations also called markov operators are investigated in the case of unital quantum channels compact operators belonging to poisson boundaries are characterized using the characterization of amenable groups by the injectivity of their von neumann algebras we will answer negatively some conjectures appearing in the work of arias et al fixed points of quantum operations j math phys 43 5872 2002 about injectivity of the commuting algebra of the kraus operators of unital quantum operations and their injective envelopes

fixed points of commutative luders operations this paper verifies a conjecture posed in a pair of papers on the fixed point sets for a class of quantum operations specifically it is proved that if a quantum operation has mutually commuting operation elements that are effects forming a resolution of the identity then the fixed point set of the quantum operation is exactly the commutant of the operation elements

on fixed points of luders operation in this paper we give a concrete example of a luders operation la with n 3 such that la b b does not imply that b commutes with all e1 e2 and e3 in a this example answers an open problem of professor gudder

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

perturbation of resonances in quantum mechanics schrodinger operators on l2 r3 of the form v with potentials v real analytic in are discussed the analytic structure in v and k with k2 the energy variable of the resolvent kernel the eigenvalues and resonances is exhibited and we obtain in particular convergent perturbation expansions for the resonances and the corresponding resonance functions the lower order expansion coefficients are computed explicitly the resonances and the corresponding functions are also computed for a particle moving under the action of n point interactions this gives asymptotic low energy information about schrodinger hamiltonians with short range potentials the perturbation theory of resonances eigenvalues and of the corresponding functions for hamiltonians describing n point interactions perturbed by a potential is also given

fixed points of quantum operations quantum operations frequently occur in quantum measurement theory quantum probability quantum computation and quantum information theory if an operator a is invariant under a quantum operation we call a a fixed point physically the fixed points are the operators that are not disturbed by the action of our main purpose is to answer the following question if a is a fixed point is a compatible with the operation elements of we shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von neumann algebras

on spectra of luders operations we show that all the eigenvalues of certain generalized luders operations are non negative real numbers in two cases of interest in particular given a commuting n tuple a a1 an consisting of positive operators on a hilbert space h satisfying j 1naj i we show that the spectrum of the luders operation a b h x j 1naj1 2xaj1 2 b h is contained in 0 so the only solution of the equation a x i x is the expected one x 12i

decoherence free linear quantum subsystems this paper provides a general theory for characterizing and constructing a decoherence free df subsystem for an infinite dimensional linear open quantum system the main idea is that based on the heisenberg picture of the dynamics rather than the commonly taken schrodinger picture the notions of controllability and observability in control theory are employed to characterize a df subsystem a particularly useful result is a general if and only if condition for a linear system to have a df component this condition is used to demonstrate how to actually construct a df dynamics in some specific examples it is also shown that as in the finite dimensional case we are able to do coherent manipulation and preservation of a state of a df subsystem

non disturbance for fuzzy quantum measurements we consider three criteria for describing non disturbance between quantum measurements while previous discussions of these criteria considered sharp measurements we shall treat more general measurements that may be unsharp or fuzzy it has been shown that in the sharp case the three criteria are equivalent to compatibility of the measurements we shall show that only the third criterion is equivalent to compatibility for the general fuzzy case moreover the first two criteria are not symmetric relative to the two measurements

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

fixed points of dual quantum operations abstract let a be a normal completely positive map on b h with kraus operators a i i 1 denote m the subset of normal completely positive maps by m a i 1 a i a i i i 1 a i a i i and a is normal in this note the relations between the fixed points of a and a are investigated we obtain that b k h a b b b k h a b b where k h is the set of all compact operators on h and a is the dual of a m in addition we show that the map a a is a bijection on m

second derivative methods applied to implicit first and second order systems a wide variety of methods have been proposed for solving implicit second order systems of differential equations such as arise in structural and mechanical vibration analyses here we discuss a class of methods based on second derivative formulae some of which have previously been used by enright 1 we show them to have desirable analytic and implementation properties we relate these methods to others recently proposed by brusa and nigro and by serbin showing that we obtain algorithms which have all the desirable properties of their methods and which can also be extended to classes of practical problems which were not considered by these other authors since implicit first order systems are also important in practice for example in finite element analyses of heat conduction problems we demonstrate the efficient implementation of the second derivative formulae in this case too

the range of generalized quantum operations abstract a completely positive map is called a generalized quantum operation if tr a tr a for all positive operators a in this note we mainly characterize the sets of b is a generalized quantum operation b is a generalized quantum operation and b and are generalized quantum operations where is a dual of and b is a self adjoint operator in particular the third set is relevant to uhlmann s theorem in quantum information

multichannel nonlinear scattering for nonintegrable equations ii the case of anisotropic potentials and data abstract the nonlinear scattering and stability results of soffer and weinstein comm math phys 1990 are extended to the case of anisotropic potentials and data the range of nonlinearities for which the theory is shown to be valid is also extended considerably

asymptotic expansions in time for solutions of schrodinger type equations let ia i p d v be a dissipative operator in l2 rn where p d is an elliptic differential operator of order m with real constant coefficients and v is a compact operator from the weighted sobolev space hm s rn to h0 p s rn s r for some m 1 let r z be the resolvent of a then an asymptotic expansion of r z as z approaches a critical value of the polynomial p is given the coefficient operators in the expansion are computed explicitly by using the resolvent expansion and the results of m murata 9 an asymptotic expansion of e ita as t is given

damping and phase analysis for some methods for solving second order ordinary differential equations we consider numerical methods for initial value problems for second order systems of ordinary differential equations analysing them by applying them to the test equation we discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property we also use a slightly more general test equation to derive conditions which ensure that the numerical forced oscillation is in phase with the true forced oscillation to illustrate the theory we consider the damping and phase properties of some frequently used methods

some characterizations of quantum channel in infinite hilbert spaces we first show that for any quantum states on h and on k there exists a quantum channel such that where h and k are finite or infinite dimensional hilbert spaces then we consider some conclusions for the quantum channel such that and ih exists or ih ik

nondisturbing quantum measurements we consider pairs of discrete quantum observables povms and analyze the relation between the notions of nondisturbance joint measurability and commutativity we specify conditions under which these properties coincide or differ depending for instance on the interplay between the number of outcomes and the hilbert space dimension or on algebraic properties of the effect operators we also show that non disturbance is in general not a symmetric relation and that it can be decided and quantified by means of a semidefinite program

quantum channels arising from abstract harmonic analysis we present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups g beginning with a representation of the measure algebra m g we unify and elaborate on recent counter examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions as well as present an application to the noiseless subsystems method of quantum error correction using a representation of the completely bounded fourier multiplier algebra mcba g we provide a new class of counter examples to the recently solved asymptotic quantum birkhoff conjecture along with a systematic method of producing the examples using a geometric representation of schur maps further properties of our channels including duality quantum capacity and entanglement preservation are discussed along with potential applications to additivity conjectures

state convertibility in the von neumann algebra framework we establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi finite von neumann algebras namely we establish a generalisation to this setting of nielsen s theorem on the convertibility of quantum states under local operations and classical communication locc schemes along the way we introduce an appropriate generalisation of locc operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von neumann algebras as an application of our result in the setting of mathrm ii 1 factors we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of vidal thus yielding a new way to quantify entanglement in that context building on previous work in the infinite dimensional setting we show that trace vectors play the role of maximally entangled states for general mathrm ii 1 factors examples are drawn from infinite spin chains quasi free representations of the car and discretised versions of the ccr

probability measures on groups and some related ideals in group algebras abstract let g be a locally compact group random walks on g some factorization problems in l 1 g and the significance for these of the amenability of g are studied these topics are linked in this paper via ideals in l 1 g of the form l 1 g e where is a probability measure on g cohen s factorization theorem and some ideas from ergodic theory play an important part

weighted energy decay for 1d wave equation we obtain a dispersive long time decay in weighted energy norms for solutions to the 1d wave equation with generic potential the decay extends the results obtained by murata for the 1d schrodinger equation

fixed points of normal completely positive maps on b h abstract given a sequence of bounded operators a j on a hilbert space h with j 1 a j a j 1 j 1 a j a j we study the map defined on b h by x j 1 a j x a j and its restriction to the hilbert schmidt class c 2 h in the case when the sum j 1 a j a j is norm convergent we show in particular that the operator 1 is not invertible if and only if the c algebra a generated by a j j 1 has an amenable trace this is used to show that may have fixed points in b h which are not in the commutant a of a even in the case when the weak closure of a is injective however if a is abelian then all fixed points of are in a even if the operators a j are not positive

characterizations of fixed points of quantum operations let a be a general quantum operation an operator b is said to be a fixed point of a if a b b in this note we shall show conditions under which b a fixed point a implies that b is compatible with the operation element of a in particular we offer an extension of the generalized luders theorem

private algebras in quantum information and infinite dimensional complementarity we introduce a generalized framework for private quantum codes using von neumann algebras and the structure of commutants this leads naturally to a more general notion of complementary channel which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite dimensional settings linear bosonic channels are considered and specific examples of gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem

normality and fixed points associated to commutative row contractions abstract let a a k k 1 n n is a positive integer or be a commutative row contraction on a complex hilbert space h and a the normal completely positive map associated with a we give some characterizations for a to be a normal sequence in the case that a is unital we show a is normal if either a is contained in a finite von neumann algebra or the set k h of all compact operators or k 1 n a k a k i moreover the fixed point set b h a of a is considered when a j i is convergent to a projection in strong operator topology

weighted decay estimates for the wave equation abstract in this work we study weighted sobolev spaces in r n generated by the lie algebra of vector fields 1 x 2 1 2 x j j 1 n interpolation properties and sobolev embeddings are obtained on the basis of a suitable localization in r n as an application we derive weighted l q estimates for the solution of the homogeneous wave equation for the inhomogeneous wave equation we generalize the weighted strichartz estimate established by v georgiev 1997 amer j math 119 1291 1319 and establish global existence results for the supercritical semilinear wave equation with non compact small initial data in these weighted sobolev spaces

a note on operator probability theory involving numerical ranges abstract in this note the relationships between the expectation and variance in operator probability theory and numerical range of operators are considered

fixed points for luders operations and commutators let b h be the algebra of all bounded linear operators on a complex hilbert space h and let tj j 1 b h be a sequence of operators such that j 1tj tj 1h in the weak operator topology let us define t b h b h by setting t y j 1t jytj for y b h we show that if x b h is an operator such that all the commutators xtj tjx as well as x itself are fixed points for t then xtj tjx 0 for all j 1 in the case when tj j 1 consists of commuting normal operators this shows that every fixed point for t commutes with all tj for j 1

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

toeplitz operators associated to commuting row contractions let h be a complex hilbert space and let tn n 1 be a sequence of commuting bounded operators on h such that n 1tntn ih let f t denote the space of all operators x in b h for which n 1tnxtn x and suppose that f t 0 we will show that there exists a triple k un n 1 where k is a hilbert space k h is a bounded operator and un n 1 b k is a sequence of commuting normal operators with n 1unun ik such that tn un for n 1 and for which the mapping y y is a complete isometry from the commutant of un n 1 onto the space f t moreover we show that the inverse of this mapping can be extended to a homomorphism c ih f t un n 1 from the unital c algebra generated by f t onto the commutant of un n 1 we also show that there exists a homomorphism c ih tn n 1 c ik un n 1 such that tn un for n 1 in the particular case when tn n 1 has only a finite number of non zero components it turns out that un n 1 is unitarily equivalent to the spherical unitary part of the standard commuting dilation of tn n 1

a noether theorem for stochastic operators on schatten classes abstract we show that a stochastic markov operator s acting on a schatten class c 1 satisfies the noether condition i e s a a and s a 2 a 2 where a c is a hermitian and bounded operator on a fixed separable and complex hilbert space h if and only if s e a g x e a g e a g s x e a g for any state x c 1 and all borel sets g r where e a g denotes the orthogonal projection coming from the spectral resolution a a z e a d z similar results are obtained for stochastic one parameter continuous semigroups

fixed points of commutative super operators in this paper we study fixed points of commutative super operators first under some circumstances we discuss the question of necessary and sufficient conditions for the set of fixed points of a super operator to be non trivial then we establish a structure lemma to completely characterize this set using this lemma we show that if a commutative super operator is also a generalized luders operator then its fixed points are precisely those operators intertwining the two sequences of operation elements associated with it specifically the main results of weihua and junde 2010 j phys a math theor 43 395206 are special cases of our results

unital quantum channels convex structure and revivals of birkhoff s theorem the set of doubly stochastic quantum channels and its subset of mixtures of unitaries are investigated we provide a detailed analysis of their structure together with computable criteria for the separation of the two sets when applied to o d covariant channels this leads to a complete characterization and reveals a remarkable feature instances of channels which are not in the convex hull of unitaries can become elements of this set by either taking two copies of them or supplementing with a completely depolarizing channel these scenarios imply that a channel whose noise initially resists any environment assisted attempt of correction can become perfectly correctable

an l stable rosenbrock method for step by step time integration in structural dynamics abstract a single step two stage l stable rosenbrock method is given in a form suitable for step by step integration of linear and non linear structural dynamics problems the method has second order accuracy and is a true single step method in that it does not require initial values of acceleration for each time step the method requires a single formation of the tangential stiffness matrix a single lu decomposition and two linear system substitutions but does not require iteration analysis and numerical examples are used to show the good stability and accuracy properties of the method and the importance of l stability when integrating numerically stiff problems in structural dynamics

asymptotic properties of quantum markov chains the asymptotic dynamics of discrete quantum markov chains generated by the most general physically relevant quantum operations is investigated it is shown that it is confined to an attractor space in which the resulting quantum markov chain is diagonalizable a construction procedure of a basis of this attractor space and its associated dual basis of 1 forms is presented it is applicable whenever a strictly positive quantum state exists which is contracted or left invariant by the generating quantum operation moreover algebraic relations between the attractor space and kraus operators involved in the definition of a quantum markov chain are derived this construction is not only expected to offer significant computational advantages in cases in which the dimension of the hilbert space is large and the dimension of the attractor space is small but it also sheds new light onto the relation between the asymptotic dynamics of discrete quantum markov chains and fixed points of their generating quantum operations finally we show that without any restriction our construction applies to all initial states whose support belongs to the so called recurrent subspace