rearrangement clustering pitfalls remedies and applications given a matrix of values in which the rows correspond to objects and the columns correspond to features of the objects rearrangement clustering is the problem of rearranging the rows of the matrix such that the sum of the similarities between adjacent rows is maximized referred to by various names and reinvented several times this clustering technique has been extensively used in many fields over the last three decades in this paper we point out two critical pitfalls that have been previously overlooked the first pitfall is deleterious when rearrangement clustering is applied to objects that form natural clusters the second concerns a similarity metric that is commonly used we present an algorithm that overcomes these pitfalls this algorithm is based on a variation of the traveling salesman problem it offers an extra benefit as it automatically determines cluster boundaries using this algorithm we optimally solve four benchmark problems and a 2 467 gene expression data clustering problem as expected our new algorithm identifies better clusters than those found by previous approaches in all five cases overall our results demonstrate the benefits of rectifying the pitfalls and exemplify the usefulness of this clustering technique our code is available at our websites

multiplicativity of completely bounded p norms implies a strong converse for entanglement assisted capacity the fully quantum reverse shannon theorem establishes the optimal rate of noiseless classical communication required for simulating the action of many instances of a noisy quantum channel on an arbitrary input state while also allowing for an arbitrary amount of shared entanglement of an arbitrary form turning this theorem around establishes a strong converse for the entanglement assisted classical capacity of any quantum channel this paper proves the strong converse for entanglement assisted capacity by a completely different approach and identifies a bound on the strong converse exponent for this task namely we exploit the recent entanglement assisted meta converse theorem of matthews and wehner several properties of the recently established sandwiched renyi relative entropy also referred to as the quantum renyi divergence and the multiplicativity of completely bounded p norms due to devetak et al the proof here demonstrates the extent to which the arimoto approach can be helpful in proving strong converse theorems it provides an operational relevance for the multiplicativity result of devetak et al and it adds to the growing body of evidence that the sandwiched renyi relative entropy is the correct quantum generalization of the classical concept for all 1

impossibility of local state transformation via hypercontractivity local state transformation is the problem of transforming an arbitrary number of copies of a bipartite resource state to a bipartite target state under local operations that is given two bipartite states is it possible to transform an arbitrary number of copies of one of them into one copy of the other state under local operations only this problem is a hard one in general since we assume that the number of copies of the resource state is arbitrarily large in this paper we prove some bounds on this problem using the hypercontractivity properties of some super operators corresponding to bipartite states we measure hypercontractivity in terms of both the usual super operator norms as well as completely bounded norms

structure of states which satisfy strong subadditivity of quantum entropy with equality we give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von neumann entropy by combining a result of petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by koashi and imoto we show that such states will have the form of a so called short quantum markov chain which in turn implies that two of the systems are independent conditioned on the third in a physically meaningful sense this characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the holevo bound on accessible information

quantum reverse hypercontractivity its tensorization and application to strong converses in this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log sobolev inequalities next we prove a generalization of the stroock varopoulos inequality in the non commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2 log sobolev and 1 log sobolev inequalities respectively we then prove some tensorization type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers finally as an application of these results we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting we prove strong converse bounds for the problems of quantum hypothesis testing and classical quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive

error estimates for regularization methods in hilbert scales in this paper we study regularization methods to reconstruct the solution x of the linear ill posed problem ax y a x to y from noisy data y delta in y the regularization methods are of the general form x alpha delta bar x g alpha b s a a b s a y delta a bar x where b denotes an unbounded self adjoint strictly positive definite operator in the hilbert space x assuming ax sim x a and x bar x p le e for some bar x in x a ge 0 and p ge 0 x r b r 2 x is the norm in a hilbert scale x r r in rnus we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g alpha lambda the regularization parameter alpha and the parameter s are chosen properly

continuity of quantum channel capacities we prove that a broad array of capacities of a quantum channel are continuous that is two channels that are close with respect to the diamond norm have correspondingly similar communication capabilities we first show that the classical capacity quantum capacity and private classical capacity are continuous with the variation on arguments varepsilon apart bounded by a simple function of varepsilon and the channel s output dimension our main tool is an upper bound of the variation of output entropies of many copies of two nearby channels given the same initial state the bound is linear in the number of copies our second proof is concerned with the quantum capacities in the presence of free backward or two way public classical communication these capacities are proved continuous on the interior of the set of non zero capacity channels by considering mutual simulation between similar channels

hastings s additivity counterexample via dvoretzky s theorem the goal of this note is to show that hastings s counterexample to the additivity of minimal output von neumann entropy can be readily deduced from a sharp version of dvoretzky s theorem

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

further convergence results on the general iteratively regularized gauss newton methods under the discrepancy principle we consider the general iteratively regularized gauss newton methods xk 1 x0 g k f xk f xk f xk f xk y f xk xk x0 for solving nonlinear inverse problems f x y using the only available noise y of y satisfying y y with a given small noise level 0 in order to produce reasonable approximation to the sought solution we terminate the iteration by the discrepancy principle under much weaker conditions we derive some further convergence results which improve the existing ones and thus expand the applied range

a matrix convexity approach to some celebrated quantum inequalities abstract some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the hansen pedersen jensen inequalities a general argument is given using matrix perspectives of operator convex functions a matrix analogue of marechal s extended perspectives provides additional inequalities including a p q 1 result of lieb

a spectral algorithm for seriation and the consecutive ones problem in applications ranging from dna sequencing through archeological dating to sparse matrix reordering a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other that is given a correlation function f reflecting the desire for each pair of elements to be near each other find all permutations pi with the property that if pi i pi j pi k then f i j ge f i k and f j k ge f i k this seriation problem is a generalization of the well studied consecutive ones problem we present a spectral algorithm for this problem that has a number of interesting features whereas most previous applications of spectral techniques provide only bounds or heuristics our result is an algorithm that correctly solves a nontrivial combinatorial problem in addition spectral methods are being successfully applied as heuristics to a variety of sequencing problems and our result helps explain and justify these applications

proximity based frameworks for generating embeddings from multi output data this paper is about supervised and semi supervised dimensionality reduction dr by generating spectral embeddings from multi output data based on the pairwise proximity information two flexible and generic frameworks are proposed to achieve supervised dr sdr for multilabel classification one is able to extend any existing single label sdr to multilabel via sample duplication referred to as mesd the other is a multilabel design framework that tackles the sdr problem by computing weight proximity matrices based on simultaneous feature and label information referred to as mope as a generalization of many current techniques a diverse set of different schemes for label based proximity calculation as well as a mechanism for combining label based and feature based weight information by considering information importance and prioritization are proposed for mope additionally we summarize many current spectral methods for unsupervised dr udr single multilabel sdr and semi supervised dr ssdr and express them under a common template representation as a general guide to researchers in the field we also propose a general framework for achieving ssdr by combining existing sdr and udr models and also a procedure of reducing the computational cost via learning with a target set of relation features the effectiveness of our proposed methodologies is demonstrated with experiments with document collections for multilabel text categorization from the natural language processing domain

towards a general convergence theory for inexact newton regularizations we develop a general convergence analysis for a class of inexact newton type regularizations for stably solving nonlinear ill posed problems each of the methods under consideration consists of two components the outer newton iteration and an inner regularization scheme which applied to the linearized system provides the update in this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including landweber and steepest decent iterations iterated tikhonov method and method of conjugate gradients

private capacity of quantum channels is not additive recently there has been considerable activity on the subject of the additivity of various quantum channel capacities here we construct a family of channels with a sharply bounded classical and hence private capacity on the other hand their quantum capacity when combined with a zero private and zero quantum capacity erasure channel becomes larger than the previous classical capacity as a consequence we can conclude for the first time that the classical private capacity is nonadditive in fact in our construction even the quantum capacity of the tensor product of two channels can be greater than the sum of their individual classical private capacities we show that this violation occurs quite generically every channel can be embedded into our construction and a violation occurs whenever the given channel has a larger entanglement assisted quantum capacity than unassisted classical capacity

entropy production of doubly stochastic quantum channels we study the entropy increase of quantum systems evolving under primitive doubly stochastic markovian noise and thus converging to the maximally mixed state this entropy increase can be quantified by a logarithmic sobolev constant of the liouvillian generating the noise we prove a universal lower bound on this constant that stays invariant under taking tensor powers our methods involve a new comparison method to relate logarithmic sobolev constants of different liouvillians and a technique to compute logarithmic sobolev inequalities of liouvillians with eigenvectors forming a projective representation of a finite abelian group our bounds improve upon similar results established before and as an application we prove an upper bound on continuous time quantum capacities in the last part of this work we study entropy production estimates of discrete time doubly stochastic quantum channels by extending the framework of discrete time logarithmic sobolev inequalities to the quantum case

strong converse exponents for a quantum channel discrimination problem and quantum feedback assisted communication this paper studies the difficulty of discriminating between an arbitrary quantum channel and a replacer channel that discards its input and replaces it with a fixed state the results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination we show that in this particular setting the most general adaptive discrimination strategies provide no asymptotic advantage over non adaptive tensor power strategies this conclusion follows by proving a quantum stein s lemma for this channel discrimination setting showing that a constant bound on the type i error leads to the type ii error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels the strong converse part of the lemma states that any attempt to make the type ii error decay to zero at a rate faster than the channel relative entropy implies that the type i error necessarily converges to one we then refine this latter result by identifying the optimal strong converse exponent for this task as a consequence of these results we can establish a strong converse theorem for the quantum feedback assisted capacity of a channel sharpening a result due to bowen furthermore our channel discrimination result demonstrates the asymptotic optimality of a non adaptive tensor power strategy in the setting of quantum illumination as was used in prior work on the topic the sandwiched renyi relative entropy is a key tool in our analysis finally by combining our results with recent results of hayashi and tomamichel we find a novel operational interpretation of the mutual information of a quantum channel mathcal n as the optimal type ii error exponent when discriminating between a large number of independent instances of mathcal n and an arbitrary worst case replacer channel chosen from the set of all replacer channels

newton lavrentiev regularization for ill posed operator equations in hilbert scales in this paper we consider the two step method for approximately solving the ill posed operator equation f x f where f d f x x is a nonlinear monotone operator defined on a real hilbert space x in the setting of hilbert scales we derive the error estimates by selecting the regularization parameter a according to the adaptive method considered by pereverzev and schock in 2005 when the available data is f d with f f email protected x is of optimal order

remarks on additivity of the holevo channel capacity and of the entanglement of formation the purpose of this article is to discuss the relation between the additivity questions regarding the quantities holevo capacityofaquantumchannelt and entanglementofformationofabipartitestate in particular using the stinespring dilation theorem we give a formula for the channel capacity involving entanglement of formation this can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels we demonstrate this connection for some families of channels allowing us to calculate the entanglement cost for many states including some where a strictly smaller upper bound on the distillable entanglement is known group symmetry is used for more sophisticated analysis giving formulas valid for a class of channels this is presented in a general framework extending recent findings of vidal dur and cirac we also discuss the property of superadditivity of the entanglement of formation which would imply both the general additivity of this function under tensor products and of the holevo capacity with or without linear cost constraints

hypercontractivity for semigroups of unital qubit channels hypercontractivity is proved for products of qubit channels that belong to self adjoint semigroups the hypercontractive bound gives necessary and sufficient conditions for a product of the form e t 1 h 1 otimes cdots otimes e t n h n to be a contraction from l p to l q where l p is the algebra of 2 n dimensional matrices equipped with the normalized schatten norm and each generator h j is a self adjoint positive semidefinite operator on the algebra of 2 dimensional matrices as a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels

on landweber iteration for nonlinear ill posed problems in hilbert scales in this paper we derive convergence rates results for landweber iteration in hilbert scales in terms of the iteration index k for exact data and in terms of the noise level delta for perturbed data these results improve the one obtained recently for landweber iteration for nonlinear ill posed problems in hilbert spaces for numerical computations we have to approximate the nonlinear operator and the infinite dimensional spaces by finite dimensional ones we also give a convergence analysis for this finite dimensional approximation the conditions needed to obtain the rates are illustrated for a nonlinear hammerstein integral equation numerical results are presented confirming the theoretical ones

inequalities for quantum entropy a review with conditions for equality this article presents self contained proofs of the strong subadditivity inequality for von neumann s quantum entropy s and some related inequalities for the quantum relative entropy most notably its convexity and its monotonicity under stochastic maps moreover the approach presented here which is based on klein s inequality and lieb s theorem that the function a tr ek log a is concave allows one to obtain conditions for equality in the case of strong subadditivity which states that s 123 s 2 s 12 s 23 where the subscripts denote subsystems of a composite system equality holds if and only if log 123 log 12 log 2 log 23 using the fact that the holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy we show that equality can be attained for that bound only when the states in the ensemble commute the article concludes with an appendix giving a short description of epstein s elegant proof of lieb s the

inexact newton regularization methods in hilbert scales we consider a class of inexact newton regularization methods for solving nonlinear inverse problems in hilbert scales under certain conditions we obtain the order optimal convergence rate result

on the order optimality of the regularization via inexact newton iterations inexact newton regularization methods have been proposed by hanke and rieder for solving nonlinear ill posed inverse problems every such a method consists of two components an outer newton iteration and an inner scheme providing increments by regularizing local linearized equations the method is terminated by a discrepancy principle in this paper we consider the inexact newton regularization methods with the inner scheme defined by landweber iteration the implicit iteration the asymptotic regularization and tikhonov regularization under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of rieder we in fact obtain a more general order optimality result by considering these inexact newton methods in hilbert scales

hypercontractivity and the logarithmic sobolev inequality for the completely bounded norm we develop the notions of hypercontractivity hc and the log sobolev ls inequality for completely bounded norms of one parameter semigroups of super operators acting on matrix algebras we prove the equivalence of the completely bounded versions of hc and ls under suitable hypotheses we also prove a version of the gross lemma which allows ls at general q to be deduced from ls at q 2

a relation between completely bounded norms and conjugate channels we show a relation between a quantum channel and its conjugate c which implies that the p p schatten norm of the channel is the same as the 1 p completely bounded norm of the conjugate this relation is used to give an alternative proof of the multiplicativity of both norms

capacity bounds via operator space methods we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

minimal entropy of states emerging from noisy quantum channels we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel which is modeled by the action of a completely positive trace preserving or stochastic map we provide strong support for the conjecture that this minimal entropy is additive namely that the minimum entropy can be achieved when product states are transmitted explicitly we prove that for a tensor product of two unital stochastic maps on qubit states using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states we give a separate argument based on the geometry of the image of the set of density matrices under stochastic maps which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps we also show that the maximal norm of the output states is multiplicative for most product maps on n qubit states including all those for which at least one map is unital for the class of unital channels on c sup 2 we show that additivity of minimal entropy implies that the holevo see ieee trans inform theory vol 44 p 269 73 1998 and russian math surv p 1295 1331 1999 capacity of the channel is additive over two inputs achievable with orthogonal states and equal to the shannon capacity this implies that superadditivity of the capacity is possible only for nonunital channels

potential capacities of quantum channels we introduce potential capacities of quantum channels in an operational way and provide upper bounds for these quantities which quantify the ultimate limit of usefulness of a channel for a given task in the best possible context unfortunately except for a few isolated cases potential capacities seem to be as hard to compute as their plain analogues we thus study upper bounds on some potential capacities for the classical capacity we give an upper bound in terms of the entanglement of formation to establish a bound for the quantum and private capacity we first lift the channel to a hadamard channel and then prove that the quantum and private capacity of a hadamard channel is strongly additive implying that for these channels potential and plain capacity are equal employing these upper bounds we show that if a channel is noisy however close it is to the noiseless channel then it cannot be activated into the noiseless channel by any other contextual channel this conclusion holds for all the three capacities we also discuss the so called environment assisted quantum capacity because we are able to characterize its potential version

a new measure for analyzing and fusing sequences of objects this work is related to the combinatorial data analysis problem of seriation used for data visualization and exploratory analysis seriation re sequences the data so that more similar samples or objects appear closer together whereas dissimilar ones are further apart despite the large number of current algorithms to realize such re sequencing there has not been a systematic way for analyzing the resulting sequences comparing them or fusing them to obtain a single unifying one we propose a new positional proximity measure that evaluates the similarity of two arbitrary sequences based on their agreement on pairwise positional information of the sequenced objects furthermore we present various statistical properties of this measure as well as its normalized version modeled as an instance of the generalized correlation coefficient based on this measure we define a new procedure for consensus seriation that fuses multiple arbitrary sequences based on a quadratic assignment problem formulation and an efficient way of approximating its solution we also derive theoretical links with other permutation distance functions and present their associated combinatorial optimization forms for consensus tasks the utility of the proposed contributions is demonstrated through the comparison and fusion of multiple seriation algorithms we have implemented using many real world datasets from different application domains

inequalities for trace norms of 2 2 block matrices this paper derives an inequality relating the p norm of a positive 2 2 block matrix to the p norm of the 2 2 matrix obtained by replacing each block by its p norm the inequality had been known for integer values of p so the main contribution here is the extension to all values p 1 in a special case the result reproduces hanner s inequality a weaker inequality which applies also to non positive matrices is presented as an application in quantum information theory the inequality is used to obtain some results concerning maximal p norms of product channels

self duality for the haagerup tensor product and hilbert space factorizations d blecher and v paulsen showed that the haagerup tensor product v h w for operator spaces v and w preserves inclusions it is proved to also preserve complete quotient maps and to be self dual in the sense that it induces the haagerup norm on the algebraic tensor product v w the full operator dual space v h w is computed it coincides with the natural operator space gg2 v w of maps v w which have completely bounded factorizations through hilbert spaces with vectors identified with row matrices more generally one has the natural complete isometry gg2 v h w x gg2 v gg2 w x given hilbert spaces h and k with vectors regarded as column matrices it is shown that one may identify the operator spaces b h k and cb h k

constructive counterexamples to the additivity of the minimum output renyi entropy of quantum channels for all p 2 we present a constructive example of the violation of the additivity of minimum output renyi entropy for each p 2 the example is provided by an antisymmetric subspace of a suitable dimension we discuss the possibility of extension of the result to go beyond p 2 and obtain additivity for p 0 for a class of entanglement breaking channels

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

multiplicativity of completely bounded p norms implies a new additivity result we prove additivity of the minimal conditional entropy associated with a quantum channel represented by a completely positive cp trace preserving map when the infimum of s 12 s 1 is restricted to states of the form mathcal i otimes phi left psi rangle langle psi right we show that this follows from multiplicativity of the completely bounded norm of considered as a map from l1 lp for lp spaces defined by the schatten p norm on matrices and give another proof based on entropy inequalities several related multiplicativity results are discussed and proved in particular we show that both the usual l1 lp norm of a cp map and the corresponding completely bounded norm are achieved for positive semi definite matrices physical interpretations are considered and a new proof of strong subadditivity is presented

unbounded violation of tripartite bell inequalities we prove that there are tripartite quantum states constructed from random unitaries that can lead to arbitrarily large violations of bell inequalities for dichotomic observables as a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model this is in sharp contrast with the bipartite case where all violations are bounded by grothendieck s constant we will discuss the possibility of determining the hilbert space dimension from the obtained violation and comment on implications for communication complexity theory moreover we show that the violation obtained from generalized greenberger horne zeilinger ghz states is always bounded so that in contrast to many other contexts ghz states do not lead to extremal quantum correlations in this case in order to derive all these physical consequences we will have to obtain new mathematical results in the theories of operator spaces and tensor norms in particular we will prove the existence of bounded but not completely bounded trilinear forms from commutative c algebras finally we will relate the existence of diagonal states leading to unbounded violations with a long standing open problem in the context of banach algebras

on a regularized levenberg marquardt method for solving nonlinear inverse problems we consider a regularized levenberg marquardt method for solving nonlinear ill posed inverse problems we use the discrepancy principle to terminate the iteration under certain conditions we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source wise representations

hypercontractivity of quasi free quantum semigroups hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate a logarithmic sobolev inequality and the corresponding logarithmic sobolev constant can be inferred from the semigroup s hypercontractive norm bound we consider completely positive quantum mechanical semigroups described by a lindblad master equation to prove the norm bound we follow an approach which has its roots in the study of classical rate equations we use interpolation theorems for non commutative spaces to obtain a general hypercontractive inequality from a particular norm bound then we derive a bound on the norm from an analysis of the block diagonal structure of the semigroup s spectrum we show that the dynamics of an n qubit graph state hamiltonian weakly coupled to a thermal environment is hypercontractive as a consequence this allows for the efficient preparation of graph states in time by coupling at sufficiently low temperature furthermore we extend our results to gapped liouvillians arising from a weak linear coupling of a free fermion systems

distance based and ad hoc consensus models in ordinal preference ranking abstract this paper examines the problem of aggregating ordinal preferences on a set of alternatives into a consensus this problem has been the subject of study for more than two centuries and many procedures have been developed to create a compromise or consensus we examine a variety of structures for preference specification and in each case review the related models for deriving a consensus two classes of consensus models are discussed namely ad hoc methods evolving primarily from parliamentary settings over the past 200 years and distance or axiomatic based methods we demonstrate the levels of complexity of the various distance based models by presenting the related mathematical programming formulations for them we also present conditions for equivalence that is for yielding the same consensus ranking for some of the methods finally we discuss various extensions of the basic ordinal ranking structures paying specific attention to partial ranking voting member weighted consensus ranking with intensity of preference and rank correlation methods as alternative approaches to deriving a consensus suggestions for future research directions are given

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

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

a note on the p q norms of 2 positive maps abstract king and ruskai asked whether the p q norm of a completely positive map acting between schatten p and q classes of hermitian operators p q sup a a a q a p is equal to the p q norm of that map when acting between schatten classes of general not necessarily hermitian operators the first proof of this statement has been given by watrous here we give an alternative proof that is also valid for 2 positive maps

entanglement assisted capacity of a quantum channel and the reverse shannon theorem the entanglement assisted classical capacity of a noisy quantum channel c sub e is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel assuming that the sender and receiver have access to the resource of shared quantum entanglement which may be used up by the communication protocol we show that the capacity c sub e is given by an expression parallel to that for the capacity of a purely classical channel i e the maximum over channel inputs spl rho of the entropy of the channel input plus the entropy of the channel output minus their joint entropy the latter being defined as the entropy of an entangled purification of spl rho after half of it has passed through the channel we calculate entanglement assisted capacities for two interesting quantum channels the qubit amplitude damping channel and the bosonic channel with amplification attenuation and gaussian noise we discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel alone or in the presence of ancillary resources such as prior entanglement in the classical analog of entanglement assisted communication communication over a discrete memoryless channel dmc between parties who share prior random information we show that one parameter is sufficient i e that in the presence of prior shared random information all dmcs of equal capacity can simulate one another with unit asymptotic efficiency

on quantum fidelities and channel capacities for the discrete memoryless quantum channel we show the equivalence of two different notions of quantum channel capacity that which uses the entanglement fidelity as its criterion for success in transmission and that which uses the minimum fidelity of pure states in a subspace of the input hilbert space as its criterion as a corollary any source with entropy rate less than the capacity may be transmitted with high entanglement fidelity we also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel this enables us to simplify a known upper bound for the channel capacity it also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity

counterexample to an additivity conjecture for output purity of quantum channels a conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel as measured by the p norm should be multiplicative with respect to the tensor product of channels we disprove this conjecture for p 4 79 the same example with p also disproves a conjecture for the multiplicativity of the injective norm of hilbert space tensor products

on convergence rates of inexact newton regularizations reginn is an algorithm of inexact newton type for the regularization of nonlinear ill posed problems inverse problems 15 1999 pp 309 327 in the present article convergence is shown under weak smoothness assumptions source conditions moreover convergence rates are established some computational illustrations support the theoretical results

potential output purity of completely positive maps the notion of potential output purity of a completely positive map is introduced as a generalization of the regularized output purity an upper bound is derived for this quantity and for several classes of maps including cq classical quantum qc quantum classical hadamard channels it is shown that potential purity does not exceed the standard output purity as an application the potential purity is used to bound the logarithmic sobolev constant of a product of depolarizing channel semigroups

multiplicativity properties of entrywise positive maps abstract multiplicativity of certain maximal p q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity cp is either augmented by or replaced by the requirement that the entries of a matrix representative of the map are non negative ep in particular for integer t multiplicativity holds for the maximal 2 2t norm of a product of two maps whenever one of the pair is ep for the maximal 1 t norm for pairs of cp maps when one of them is also ep and for the maximal 1 2t norm for the product of an ep and a 2 positive map similar results are shown in the infinite dimensional setting of convolution operators on l 2 r with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix these results apply in particular to gaussian bosonic channels

thermodynamic capacity of quantum processes thermodynamics imposes restrictions on what state transformations are possible in the macroscopic limit of asymptotically many independent copies of a state as for instance in the case of an ideal gas the possible transformations become reversible and are fully characterized by the free energy in this letter we present a thermodynamic resource theory for quantum processes that also becomes reversible in the macroscopic limit a property that is especially rare for a resource theory of quantum channels we identify a unique single letter and additive quantity the thermodynamic capacity that characterizes the thermodynamic value of a quantum channel in the sense that the work required to simulate many repetitions of a quantum process employing many repetitions of another quantum process becomes equal to the difference of the respective thermodynamic capacities on a technical level we provide asymptotically optimal constructions of universal implementations of quantum processes a challenging aspect of this construction is the apparent necessity to coherently combine thermal engines that would run in different thermodynamic regimes depending on the input state our results have applications in quantum shannon theory by providing a generalized notion of quantum typical subspaces and by giving an operational interpretation to the entropy difference of a channel

clustering ensembles models of consensus and weak partitions clustering ensembles have emerged as a powerful method for improving both the robustness as well as the stability of unsupervised classification solutions however finding a consensus clustering from multiple partitions is a difficult problem that can be approached from graph based combinatorial or statistical perspectives this study extends previous research on clustering ensembles in several respects first we introduce a unified representation for multiple clusterings and formulate the corresponding categorical clustering problem second we propose a probabilistic model of consensus using a finite mixture of multinomial distributions in a space of clusterings a combined partition is found as a solution to the corresponding maximum likelihood problem using the em algorithm third we define a new consensus function that is related to the classical intraclass variance criterion using the generalized mutual information definition finally we demonstrate the efficacy of combining partitions generated by weak clustering algorithms that use data projections and random data splits a simple explanatory model is offered for the behavior of combinations of such weak clustering components combination accuracy is analyzed as a function of several parameters that control the power and resolution of component partitions as well as the number of partitions we also analyze clustering ensembles with incomplete information and the effect of missing cluster labels on the quality of overall consensus experimental results demonstrate the effectiveness of the proposed methods on several real world data sets

recent advances in the solution of quadratic assignment problems the quadratic assignment problem qap is notoriously difficult for exact solution methods in the past few years a number of long open qaps including those posed by steinberg 1961 nugent et al 1968 and krarup 1972 were solved to optimality for the first time the solution of these problems has utilized both new algorithms and novel computing structures we describe these developments as well as recent work which is likely to result in the solution of even more difficult instances

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

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces

heralded channel holevo superadditivity bounds from entanglement monogamy we introduce a class of channels called heralded channels which generalize the quantum erasure channel composed with an arbitrary other quantum channel we show that monogamy of squashed entanglem

on converse bounds for classical communication over quantum channels we explore several new converse bounds for classical communication over quantum channels in both the one shot and asymptotic regimes first we show that the matthews wehner meta converse bound for entanglement assisted classical communication can be achieved by activated no signaling assisted codes suitably generalizing a result for classical channels second we derive a new efficiently computable meta converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel as applications we provide a finite resource analysis of classical communication over quantum erasure channels including the second order and moderate deviation asymptotics third we explore the asymptotic analogue of our new meta converse the upsilon information of the channel we show that its regularization is an upper bound on the classical capacity which is generally tighter than the entanglement assisted capacity and other known efficiently computable strong converse bounds for covariant channels we show that the upsilon information is a strong converse bound

the quantum capacity with symmetric side channels in this paper we present an upper bound for the quantum channel capacity that is both additive and convex our bound can be interpreted as the capacity of a channel for high fidelity quantum communication when assisted by a family of channels that have no capacity on their own this family of assistance channels which we call symmetric side channels consists of all channels mapping symmetrically to their output and environment the bound seems to be quite tight and for degradable quantum channels it coincides with the unassisted channel capacity using this symmetric side channel capacity we find new upper bounds on the capacity of the depolarizing channel we also briefly indicate an analogous notion for distilling entanglement using the same class of one way channels yielding one of the few entanglement measures that is monotonic under local operations with one way classical communication 1 locc but not under the more general class of local operations with classical communication locc

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

new multiplicativity results for qubit maps let be a trace preserving positivity preserving but not necessarily completely positive linear map on the algebra of complex 2 2 matrices and let be any finite dimensional completely positive map for p 2 and p 4 we prove that the maximal p norm of the product map is the product of the maximal p norms of and restricting to the class of completely positive maps this settles the multiplicativity question for all qubit channels in the range of values p 4