minimax control of nonlinear evolution equations in this paper we study nonlinear parametric optimal control problems we formulate the relevant minimax problem and prove the existence of an optimal control also we derive necessary conditions for saddle point optimality we also associate to the original optimization problem a dual problem and then for the pair we prove a complete duality theorem finally we present two examples of parabolic distributed parameter systems which illustrate the abstract results

minimax control of nonlinear evolution equations in this paper we study nonlinear parametric optimal control problems we formulate the relevant minimax problem and prove the existence of an optimal control also we derive necessary conditions for saddle point optimality we also associate to the original optimization problem a dual problem and then for the pair we prove a complete duality theorem finally we present two examples of parabolic distributed parameter systems which illustrate the abstract results

on the exponential decay of the euler bernoulli beam with boundary energy dissipation abstract we study the asymptotic behavior of the euler bernoulli beam which is clamped at one end and free at the other end we apply a boundary control with memory at the free end of the beam and prove that the exponential decay of the memory kernel is a necessary and sufficient condition for the exponential decay of the energy

properties of subentropy subentropy is an entropy like quantity that arises in quantum information theory for example it provides a tight lower bound on the accessible information for pure state ensembles dual to the von neumann entropy upper bound in holevo s theorem here we establish a series of properties of subentropy paralleling the well developed analogous theory for von neumann entropy further we show that subentropy is a lower bound for min entropy we introduce a notion of conditional subentropy and show that it can be used to provide an upper bound for the guessing probability of any classical quantum state of two qubits we conjecture that the bound applies also in higher dimensions finally we give an operational interpretation of subentropy within classical information theory

random private quantum states the study of properties of randomly chosen quantum states has in recent years led to many insights into quantum entanglement in this work we study private quantum states from this point of view private quantum states are bipartite quantum states characterised by the property that carrying out simple local measurements yields a secret bit this feature is shared by the maximally entangled pair of quantum bits yet private quantum states are more general and can in their most extreme form be almost bound entangled in this work we study the entanglement properties of random private quantum states and show that they are hardly distinguishable from separable states and thus have low repeatable key despite containing one bit of key the technical tools we develop are centred around the concept of locally restricted measurements and include a new operator ordering bounds on norms under tensoring with entangled states and a continuity bound for a relative entropy measure

relaxation in nonconvex optimal control problems described by fractional differential equations abstract we consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control a relaxation problem is treated along with the original problem it is proved that under general assumptions the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory the control and the functional in appropriate topologies simultaneously

the eigenvalue problem for networks of beams abstract in this paper we consider the spectral analysis of different models of networks of euler bernoulli beams we first give the characteristic equation for the spectrum secondly in some particular situations we show that the spectrum depends only on the structure of the graph thirdly we investigate the asymptotic behaviour of the eigenvalues by proving the so called weyl s formula

compensator design for stability enhancement with collocated controllers a continuum model rather than a finite element model is used the optimal compensator design is formulated as a stochastic regulator problem and is shown to be solvable by the general infinite dimensional theory developed by the author despite the lack of exponential stabilizability infinite dimensional steady state riccati equations characterizing the feedback control gain and the kalman filter gain operators can be solved explicitly the associated performance indexes including the mean square control effort are calculated in closed form as a first approximation the compensator transfer function can be realized as a bank of bandpass filters in parallel centered at the undamped mode frequencies numerical calculations for the gain and bandwidths for a typical configuration are presented the performance of the compensator is evaluated when in fact in the true model there is no actuator noise the theoretical problem involved is to show that the infinite dimensional stochastic process is asymptotically stationary it is possible to calculate the steady state covariance in closed form and thereby calculate performance indexes of interest explicitly facilitating the choice of optimal design parameters

on a new class of differential variational inequalities and a stability result this paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering operations research and physical sciences this new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right hand sides differential algebraic systems with constraints dynamic complementarity systems and evolutionary variational systems the purpose of this paper is two fold firstly we show that these differential variational inequalities when considering slow solutions and the more general level of a hilbert space contain projected dynamical systems another recent subclass of general differential inclusions this relation follows from a precise geometric description of the directional derivative of the metric projection in hilbert space which is based on the notion of the quasi relative interior secondly we are concerned with stability of the solution set to this class of differential variational inequalities here we present a novel upper set convergence result with respect to perturbations in the data including perturbations of the associated set valued maps and the constraint set here we impose weak convergence assumptions on the perturbed set valued maps use the monotonicity method of browder and minty and employ mosco convergence as set convergence also as a consequence we obtain a stability result for linear complementarity systems

exponential decay of correlations implies area law we prove that a finite correlation length i e exponential decay of correlations implies an area law for the entanglement entropy of quantum states defined on a line the entropy bound is exponential in the correlation length of the state thus reproducing as a particular case hastings s proof of an area law for groundstates of 1d gapped hamiltonians

entanglement of random subspaces via the hastings bound recently hastings a counterexample to additivity of minimum output entropy nat phys 5 255 2009 e print arxiv 0809 3972v3 proved the existence of random unitary channels which violate the additivity conjecture in this paper we use hastings method to derive new bounds for the entanglement of random subspaces of bipartite systems as an application we use these bounds to prove the existence of nonunital channels which violate additivity of minimal output entropy

stability of frustration free hamiltonians we prove stability of the spectral gap for gapped frustration free hamiltonians under general quasi local perturbations we present a necessary and sufficient condition for stability which we call local topological quantum order and show that this condition implies an area law for the entanglement entropy of the groundstate subspace this result extends previous work by bravyi et al on the stability of topological quantum order for hamiltonians composed of commuting projections with a common zero energy subspace we conclude with a list of open problems relevant to spectral gaps and topological quantum order

efficient approximation of quantum channel capacities we propose an iterative method for approximating the capacity of classical quantum channels with a discrete input alphabet and a finite dimensional output under additional constraints on the input distribution based on duality of convex programming we derive explicit upper and lower bounds for the capacity to provide an additive varepsilon close estimate to the capacity the presented algorithm requires o n vee m m 3 log n 1 2 varepsilon 1 steps where n denotes the input alphabet size and m denotes the output dimension we then generalize the method to the task of approximating the capacity of classical quantum channels with a bounded continuous input alphabet and a finite dimensional output this using the idea of a universal encoder allows us to approximate the holevo capacity for channels with a finite dimensional quantum mechanical input and output in particular we show that the problem of approximating the holevo capacity can be reduced to a multi dimensional integration problem for certain families of quantum channels we prove that the complexity to derive an additive varepsilon close solution to the holevo capacity is subexponential or even polynomial in the problem size we provide several examples to illustrate the performance of the approximation scheme in practice

tight uniform continuity bounds for the quantum conditional mutual information for the holevo quantity and for capacities of quantum channels we start with fannes type and winter s type tight uniform continuity bounds for the quantum conditional mutual information and their specifications for states of special types then we analyse continuity of the holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states we show that the holevo quantity is continuous on the set of all ensembles of m states with respect to all the metrics if either m or the dimension of underlying hilbert space is finite and obtain fannes type tight continuity bounds for the holevo quantity in this case in the general case conditions for local continuity of the holevo quantity for discrete and continuous ensembles are found winter s type tight continuity bound for the holevo quantity under constraint on the average energy of ensembles is obtained and applied to the system of quantum oscillators the above results are used to obtain tight and close to tight continuity bounds for basic capacities of finite dimensional chann

operational resource theory of coherence we establish an operational theory of coherence or of superposition in quantum systems by focusing on the optimal rate of performance of certain tasks namely we introduce the two basic concepts coherence distillation and coherence cost in the processing quantum states under so called incoherent operations baumgratz cramer and plenio phys rev lett 113 140401 2014 we then show that in the asymptotic limit of many copies of a state both are given by simple single letter formulas the distillable coherence is given by the relative entropy of coherence in other words we give the relative entropy of coherence its operational interpretation and the coherence cost by the coherence of formation which is an optimization over convex decompositions of the state an immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state but no coherence could be distilled from it further we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states

a sharp continuity estimate for the von neumann entropy we derive an inequality relating the entropy difference between two quantum states to their trace norm distance sharpening a well known inequality due to fannes in our inequality equality can be attained for every prescribed value of the trace norm distance

assisted work distillation we study the process of assisted work distillation this scenario arises when two parties share a bipartite quantum state ab and their task is to locally distill the optimal amount of work when one party is restricted to thermal operations whereas the other can perform general quantum operations and they are allowed to communicate classically we demonstrate that this question is intimately related to the distillation of classical and quantum correlations in particular we show that the advantage of one party performing global measurements over many copies of ab is related to the nonadditivity of the entanglement of formation we also show that there may exist work bound in the quantum correlations of the state that is only extractable under the wider class of local gibbs preserving operations

on unified entropy characterization of quantum channels we consider properties of quantum channels with the use of unified entropies extremal unravelings of quantum channel with respect to these entropies are examined the concept of map entropy is extended in terms of the unified entropies the map q s entropy is naturally defined as the unified q s entropy of a rescaled dynamical matrix of given quantum channel inequalities of fannes type are obtained for introduced entropies in terms of both the trace and frobenius norms of difference between corresponding dynamical matrices additivity properties of introduced map entropies are discussed the known inequality of lindblad with the entropy exchange is generalized to many of the unified entropies for the tensor product of a pair of quantum channels we derive a two sided estimate on the output entropy of a maximally entangled input state

maximum and minimum entropy states yielding local continuity bounds given an arbitrary quantum state we obtain an explicit construction of a state e respectively e which has the maximum respectively minimum entropy among all states which lie in a specified neighborhood e ball of computing the entropy of these states leads to a local strengthening of the continuity bound of the von neumann entropy i e the audenaert fannes inequality our bound is local in the sense that it depends on the spectrum of the states e and e depend only on the geometry of the e ball and are in fact optimizers for a larger class of entropies these include the renyi entropy and the minimum and maximum entropies providing explicit formulas for certain smoothed quantities this allows us to obtain local continuity bounds for these quantities as well in obtaining this bound we first derive a more general result which may be of independent interest namely a necessary and sufficient condition under which a state maximizes a concave

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

the quantum reverse shannon theorem based on one shot information theory the quantum reverse shannon theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel s entanglement assisted classical capacity in this paper we provide a new proof of this theorem which has previously been proved by bennett devetak harrow shor and winter our proof has a clear structure being based on two recent information theoretic results one shot quantum state merging and the post selection technique for quantum channels

on parametric evolution inclusions of the subdifferential type with applications to optimal control problems in this paper we study parametric evolution inclusions of the subdifferential type and their applications to the sensitivity analysis of nonlinear infinite dimensional optimal control problems the parameter appears in all the data of the problem including the subdifferential operator first we establish several continuity results for the solution multifunction of the subdifferential inclusion then we study how these results can be used to examine the sensitivity properties variational stability of certain broad classes of nonlinear infinite dimensional optimal control problems some examples are worked out in detail illustrating the applicability of our work these include obstacle problems with time varying obstacles optimal control of distributed parameter systems and differential variational inequalities

approximate degradable quantum channels degradable quantum channels are an important class of completely positive trace preserving maps among other properties they offer a single letter formula for the quantum and the private classical capacity and are characterized by the fact that a complementary channel can be obtained from the channel by applying a degrading channel in this paper we introduce the concept of approximate degradable channels which satisfy this condition up to some finite varepsilon geq 0 that is there exists a degrading channel which upon composition with the channel is varepsilon close in the diamond norm to the complementary channel we show that for any fixed channel the smallest such varepsilon can be efficiently determined via a semidefinite program moreover these approximate degradable channels also approximately inherit all other properties of degradable channels as an application we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication

distributed compression of correlated classical quantum sources or the price of ignorance we resume the investigation of the problem of independent local compression of correlated quantum sources the classical case of which is covered by the celebrated slepian wolf theorem we focus specifically on classical quantum cq sources for which one edge of the rate region corresponding to the compression of the classical part using the quantum part as side information at the decoder was previously determined by devetak and winter phys rev a 68 042301 2003 whereas the devetak winter protocol attains a rate sum equal to the von neumann entropy of the joint source here we show that the full rate region is much more complex due to the partially quantum nature of the source in particular in the opposite case of compressing the quantum part of the source using the classical part as side information at the decoder typically the rate sum is strictly larger than the von neumann entropy of the total source we determine the full rate region in the generic case showing that apart from the devetak winter point all other points in the achievable region have a rate sum strictly larger than the joint entropy we can interpret the difference as the price paid for the quantum encoder being ignorant of the classical side information in the general case we give an achievable rate region via protocols that are built on the decoupling principle and the protocols of quantum state merging and quantum state redistribution our achievable region is matched almost by a single letter converse which however still involves asymptotic errors and an unbounded auxiliary system

stabilization of vibrating beam with a tip mass controlled by combined feedback forces abstract a flexible structure consisting of a euler bernoulli beam with a tip mass is considered to stabilize this system we use a boundary control laws uxxx 1 t mutt 1 t ut 1 t uxxxt 1 t and uxx 1 t uxt 1 t a sensitivity asymptotic analysis of the system s eigenvalues and eigenfunctions is set up we prove that all of the generalized eigenfunctions of 2 9 form a riesz basis of h by a new method we prove that the operator a generates a c0 contraction semigroup t t t 0 furthermore t t t 0 is uniformly exponentially stable and the optimal exponential decay rate can be obtained from the spectrum of the system

multipartite quantum correlations and local recoverability characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory more recently however the total correlation or multipartite information measure has been helpful in accomplishing this goal especially with the multipartite symmetric quantum msq discord piani et al 2008 phys rev lett 100 090502 doi 10 1103 physrevlett 100 090502 1 and the conditional entanglement of multipartite information cemi yang et al 2008 phys rev lett 101 140501 doi 10 1103 physrevlett 101 140501 2 here we apply a recent and significant improvement of strong subadditivity of quantum entropy fawzi renner 2014 in order to develop these quantities further in particular we prove that the msq discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems furthermore we prove that the cemi is a faithful entanglement measure i e it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state along the way we provide an operational interpretation of the msq discord in terms of the partial state distribution protocol which in turn as a special case gives an interpretation for the original discord quantity finally we prove an inequality that could potentially improve upon the fawzi renner inequality in the multipartite context but it remains an open question to determine whether this is so 1 http dx doi org 10 1103 physrevlett 100 090502 2 http dx doi org 10 1103 physrevlett 101 140501

tight bound on relative entropy by entropy difference we prove a lower bound on the relative entropy between two finite dimensional states in terms of their entropy difference and the dimension of the underlying space the inequality is tight in the sense that equality can be attained for any prescribed value of the entropy difference both for quantum and classical systems we outline implications for information theory and thermodynamics such as a necessary condition for a process to be close to thermodynamic reversibility or an easily computable lower bound on the classical channel capacity furthermore we derive a tight upper bound uniform for all states of a given dimension on the variance of the surprisal whose thermodynamic meaning is that of heat capacity

channel capacities via p summing norms abstract in this paper we show how the metric theory of tensor products developed by grothendieck perfectly fits in the study of channel capacities a central topic in shannon s information theory furthermore in the last years shannon s theory has been fully generalized to the quantum setting and revealed qualitatively new phenomena in comparison in this paper we consider the classical capacity of quantum channels with restricted assisted entanglement these capacities include the classical capacity and the unlimited entanglement assisted classical capacity of a quantum channel our approach to restricted capacities is based on tools from functional analysis and in particular the notion of p summing maps going back to grothendieck s work pisier s noncommutative vector valued l p spaces allow us to establish the new connection between functional analysis and information theory in the quantum setting

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

multivalued perturbations of subdifferential type evolution equations in hilbert spaces in this paper we study the multivalued evolution equation x t x t f t x t x 0 x0 where x r is a proper convex lower semicontinuous l s c function f is a multivalued perturbation and x is an infinite dimensional separable hilbert space we have an existence result for f being nonconvex valued and another for f being convex valued but not closed valued when k indicator function of a compact convex set k we obtain some extensions of earlier results by moreau and henry then using the kuratowski mosco convergence of sets and the convergence of functions we prove a well posedness result for the evolution inclusion we are studying also we consider a random version of it and prove the existence of a random solution finally we present applications to problems in partial differential equations

continuity bounds on the quantum relative entropy ii the quantum relative entropy is frequently used as a distance measure between two quantum states and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory we have derived many such inequalities in previous work the present paper is a follow up on this and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned the result obtained here is more general than the corresponding one from our previous work as a corollary we obtain a sharp upper bound on the regularised relative entropy introduced by lendi farhadmotamed and van wonderen

quantum state discrimination bounds for finite sample size in the problem of quantum state discrimination one has to determine by measurements the state of a quantum system based on the a priori side information that the true state is one of the two given and completely known states or in general it is not possible to decide the identity of the true state with certainty and the optimal measurement strategy depends on whether the two possible errors mistaking for or the other way around are treated as of equal importance or not results on the quantum chernoff and hoeffding bounds and the quantum stein s lemma show that if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies and the decay rate is given by a certain statistical distance between and the chernoff distance the hoeffding distances and the relative entropy respectively while these results provide a complete solution to the asymptotic problem they are not completely satisfying from a practical point of vi

tight uniform continuity bounds for quantum entropies conditional entropy relative entropy distance and energy constraints we present a bouquet of continuity bounds for quantum entropies falling broadly into two classes first a tight analysis of the alicki fannes continuity bounds for the conditional von neumann entropy reaching almost the best possible form that depends only on the system dimension and the trace distance of the states almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators as applications we give new proofs with tighter bounds of the asymptotic continuity of the relative entropy of entanglement er and its regularization e r infty as well as of the entanglement of formation ef using a novel quantum coupling of density operators which may be of independent interest we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation aka entanglement cost e c e f infty second we derive analogous continuity bounds for the von neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint most importantly systems of multiple quantum harmonic oscillators while without an energy bound the entropy is discontinuous it is well known to be continuous on states of bounded energy however a quantitative statement to that effect seems not to have been known here under some regularity assumptions on the hamiltonian we find that quite intuitively the gibbs entropy at the given energy roughly takes the role of the hilbert space dimension in the finite dimensional fannes inequality

entanglement cost of quantum channels the entanglement cost of a quantum channel is the minimal rate at which entanglement between sender and receiver is needed in order to simulate many copies of a quantum channel in the presence of free classical communication in this paper we show how to express this quantity as a regularized optimization of the entanglement formation over states that can be generated between sender and receiver our formula is the channel analog of a well known formula for the entanglement cost of quantum states in terms of the entanglement of formation and shares a similar relation to the recently shattered hope for additivity the entanglement cost of a quantum channel can be seen as the analog of the quantum reverse shannon theorem in the case where free classical communication is allowed the techniques used in the proof of our result are then also inspired by a recent proof of the quantum reverse shannon theorem and feature the one shot formalism for quantum information theory the postselection technique for quantum channels as well as sion s minimax theorem we discuss two applications of our result first we are able to link the security in the noisy storage model to a problem of sending quantum rather than classical information through the adversary s storage device this not only improves the range of parameters where security can be shown but also allows us to prove security for storage devices for which no results were known before second our result has consequences for the study of the strong converse quantum capacity here we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity

necessary and sufficient conditions for l 1 strong weak lower semicontinuity of integral functionals soit t 7b f un espace fini de mesure nonatomique x ii ii un espace de banach non separable et v i i un espace de banach reflexif separable dont v est le dual soit f txxxv une fonction 7b f 7b b x v mesurable on etudie l integrale fonctionnelle associee i f l x 1 l v 1 definie par if x v f r x t v t dt

modeling stabilization and control of serially connected beams many flexible structures consist of a large number of components coupled end to end in the form of a chain in this paper we consider the simplest type of such structures which is formed by n seri

uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy we obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension if it is finite and on the input energy bound if the input dimension is infinite we pay special attention to the case in which a multimode quantum oscillator is an input system first we prove continuity bounds for the output conditional mutual information for a single channel and for copies of a channel then we obtain estimates for the variation of the output holevo quantity with respect to simultaneous variations of a channel and of an input ensemble as a result tight and close to tight continuity bounds for basic capacities of quantum channels which depend on the input dimension are obtained they complement the leung smith continuity bounds which depend on the output dimension finally we obtain tight and close to tight continuity bounds for basic capacities of infinite dimensional energy constrained channels with respect to the energy constrained bures distance generating the strong convergence of quantum channels

on fixed point stability for set valued contractive mappings with applications to generalized differential equations where d x c inf d x c a set valued mapping t x c x is a ci 1 and for x y e x if x is complete then every set valued contraction has a fixed point i e a point x with x e tx the set of fixed points of t will be denoted by f t stability of fixed points of set valued contractions was investigated in 12 and lo in lo a stability theorem was proved under some rather restrictive conditions including i the domain of the maps being a closed convex bounded subset of a hilbert space ii the image of each point under each map being a closed convex subset theorem 1 below removes these conditions

diagonal quantum discord quantum discord measures quantum correlation by comparing the quantum mutual information with the maximal amount of mutual information accessible to a quantum measurement this paper analyzes the properties of diagonal discord a simplified version of discord that compares quantum mutual information with the mutual information revealed by a measurement that correspond to the eigenstates of the local density matrices in contrast to the optimized discord diagonal discord is easily computable it also finds connections to thermodynamics and resource theory here we further show that for the generic case of non degenerate local density matrices diagonal discord exhibits desirable properties as a preferable discord measure we employ the theory of resource destroying maps liu z w et al 2017 phys rev lett 118 060502 to prove that diagonal discord is monotonically nonincreasing under the operation of local discord nongenerating qudit channels d 2 and provide numerical evidence that such monotonicity holds for qubit channels as well we also show that it is continuous and derive a fannes like continuity bound our results hold for a variety of simple discord measures generalized from diagonal discord