one shot coherence dilution manipulation and quantification of quantum resources are fundamental problems in quantum physics in the asymptotic limit coherence distillation and dilution have been proposed by manipulating infinite identical copies of states in the nonasymptotic setting finite data size effects emerge and the practically relevant problem of coherence manipulation using finite resources has been left open this letter establishes the one shot theory of coherence dilution which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations dephasing covariant incoherent operations incoherent operations or strictly incoherent operations we introduce several coherence monotones with concrete operational interpretations that estimate the one shot coherence cost the minimum amount of maximally coherent states needed for faithful coherence dilution furthermore we derive the asymptotic coherence dilution results with maximally incoherent operations incoherent operations and strictly incoherent operations as special cases our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources such as quantum key distribution and random number generation

limits for entanglement measures the basic principle of entanglement processing says that entanglement cannot increase under local operations and classical communication based on this principle we show that any entanglement measure e suitable for the regime of a high number of identically prepared entangled pairs satisfies ed or e or ef where ed and ef are the entanglement of distillation and formation respectively moreover we exhibit a theorem establishing a very general form of bounds for distillable entanglement

existence of a tree of stieltjes strings corresponding to two given spectra transversal vibrations of a plane rooted tree of stieltjes strings are considered with dirichlet boundary conditions at each pendant vertex continuity and kirchhoff conditions are imposed at each interior vertex except the root the dirichlet problem is that with the dirichlet condition at the root and the neumann problem is that with continuity and kirchhoff conditions at the root it is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of neumann and dirichlet problems generated by stieltjes string recurrent relations on any prescribed tree

moment based evidence for simple rational valued hilbert schmidt generic 2 2 separability probabilities employing the hilbert schmidt measure we explicitly compute and analyze a number of determinantal product bivariate moments k pt n k n 0 1 2 3 with pt denoting the partial transpose for both generic 9 dimensional two rebit and generic 15 dimensional two qubit 1 density matrices the results are then incorporated into a general formula parameterized by k n and with the case 2 presumptively corresponding to generic 27 dimensional quaternionic systems holding the dyson index like parameter fixed the induced univariate moments pt n and pt n are inputted into a legendre polynomial based least squares probability distribution reconstruction algorithm of provost 2005 mathematica j 9 727 yielding specific separability probability estimates since as the number of inputted moments grows estimates based on the variable pt strongly decrease while ones employing pt strongly increase and converge faster the gaps between upper and lower estimates diminish yielding sharper and sharper bounds remarkably for 2 with the use of 2325 moments a separability probability lower bound 0 999 999 987 as large as is found for 1 based on 2415 moments a lower bound results that is 0 999 997 066 times as large as a simpler still fractional value that had previously been conjectured slater 2007 j phys a math theor 40 14279 furthermore for employing 3310 moments the lower bound is 0 999 955 times as large as a rational value previously considered slater 2010 j phys a math theor 43 195302

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

entanglement cost for sequences of arbitrary quantum states the entanglement cost of arbitrary sequences of bipartite states is shown to be expressible as the minimization of a conditional spectral entropy rate over sequences of separable extensions of the states in the sequence the expression is shown to reduce to the regularized entanglement of formation when the nth state in the sequence consists of n copies of a single bipartite state

quantifying entanglement resources we present an overview of the quantitative theory of single copy entanglement in finite dimensional quantum systems in particular we emphasize the point of view that different entanglement measures quantify different types of resources which leads to a natural interdependence of entanglement classification and quantification apart from the theoretical basis we outline various methods for obtaining quantitative results on arbitrary mixed states this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

statistical properties of random density matrices statistical properties of ensembles of random density matrices are investigated we compute traces and von neumann entropies averaged over ensembles of random density matrices distributed according to the bures measure the eigenvalues of the random density matrices are analysed we derive the eigenvalue distribution for the bures ensemble which is shown to be broader then the quarter circle distribution characteristic of the hilbert schmidt ensemble for measures induced by partial tracing over the environment we compute exactly the two point eigenvalue correlation function

probability measure generated by the superfidelity we study the probability measure on the set of density matrices induced by the metric defined by using superfidelity we give the formula for the probability density of eigenvalues we also study some statistical properties of the set of density matrices equipped with the introduced measure and provide a method for generating density matrices according to the introduced measure

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

quantum rate distortion coding with auxiliary resources we extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression the first setting we consider is that of quantum rate distortion coding with the help of a classical side channel our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function extending earlier work of devetak and berger we also combine this bound with the entanglement assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source the second setting we consider is that of quantum rate distortion coding with quantum side information qsi available to the receiver in order to prove results in this setting we first state and prove a quantum reverse shannon theorem with qsi for tensor power states which extends the known tensor power quantum reverse shannon theorem the achievability part of this theorem relies on the quantum state redistribution protocol while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver s qsi this quantum reverse shannon theorem with qsi naturally leads to quantum rate distortion theorems with qsi with or without entanglement assistance

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

classification of joint numerical ranges of three hermitian matrices of size three abstract the joint numerical range w f of three hermitian 3 by 3 matrices f f 1 f 2 f 3 is a convex and compact subset in r 3 generically we find that w f is a three dimensional oval assuming dim w f 3 every one or two dimensional face of w f is a segment or a filled ellipse we prove that only ten configurations of these segments and ellipses are possible we identify a triple f for each class and illustrate w f using random matrices and dual varieties

all teleportation and dense coding schemes we establish a one to one correspondence between 1 quantum teleportation schemes 2 dense coding schemes 3 orthonormal bases of maximally entangled vectors 4 orthonormal bases of unitary operators with respect to the hilbert schmidt scalar product and 5 depolarizing operations whose kraus operators can be chosen to be unitary the teleportation and dense coding schemes are assumed to be tight in the sense that all hilbert spaces involved have the same finite dimension d and the classical channel involved distinguishes d 2 signals a general construction procedure for orthonormal bases of unitaries involving latin squares and complex hadamard matrices is also presented

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

entanglement cost of antisymmetric states and additivity of capacity of some quantum channels we study the entanglement cost of the states in the antisymmetric space which consists of d 1 d dimensional systems the cost is always log2 d 1 ebits when the state is divided into bipartite combined with the arguments in 6 additivity of channel capacity of some quantum channels is also shown

hilbert schmidt volume of the set of mixed quantum states we compute the volume of the convex n2 1 dimensional set n of density matrices of size n with respect to the hilbert schmidt measure the hyper area of the boundary of this set is also found and its ratio to the volume provides information about the structure of n similar investigations are also performed for the smaller set of all real density matrices as an intermediate step we analyse volumes of the unitary and orthogonal groups and of the flag manifolds

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

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

bounds on localizable information via semidefinite programming we investigate the so called localizable information of bipartite states and a parallel notion of information deficit localizable information is defined as the amount of information that can be concentrated by means of classical communication and local operations where only maximally mixed states can be added for free the information deficit is defined as difference between total information contents of the state and localizable information we consider a larger class of operations the so called ppt operations which in addition preserve maximally mixed state ppt pmm operations we formulate the related optimization problem as a semidefinite program with suitable constraints we then provide bound for fidelity of transition of a given state into product pure state on hilbert space of dimension d this allows to obtain a general upper bound for localizable information and also for information deficit we calculated the bounds exactly for werner states and isotropic states in any dimension surprisin

an inverse eigenvalue problem for pseudo jacobi matrices abstract in this paper the theory on direct and inverse spectral problems for jacobi matrices is revisited in a kind of pseudo jacobi matrices j n r with a mixed path as its graph in the non self adjoint setting in this context a sign change in one of the nondiagonal entries of the matrix yields strong perturbations in its spectral properties the reconstruction of a pseudo jacobi matrix from its spectrum and the spectra of two complementary principal matrices is investigated an algorithm for the reconstruction of matrices from prescribed spectral data is provided and illustrative numerical experiments are performed

one can hear the composition of a string experiments with an inverse eigenvalue problem to what extent do the vibrations of a mechanical system reveal its composition despite innumerable applications and mathematical elegance this question often slips through those cracks that separate courses in mechanics differential equations and linear algebra we address this omission by detailing a classical finite dimensional example the use of frequencies of vibration to recover positions and masses of beads vibrating on a string first we derive the equations of motion then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory s monochord more challenging is the recovery of masses and positions of the beads from spectral data a problem for which a variety of elegant algorithms exist after presenting one such method based on orthogonal polynomials in a manner suitable for advanced undergraduates we confirm its efficacy through physical experiment we encourage readers to conduct their own explorations using the numerous data sets we provide

free self decomposability and unimodality of the fuss catalan distributions we study properties of the fuss catalan distributions mu p r p ge 1 0 1 there exists r 0 p with p 1 r 0 p p such that the fuss catalan distribution mu p r is unimodal if and only if either r p or 0 r le r 0 p we prove rigorously some partial results

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

accessibility of physical states and non uniqueness of entanglement measure ordering physical states is the key to quantifying some physical property of the states uniquely bipartite pure entangled states are totally ordered under local operations and classical communication locc in the asymptotic limit and uniquely quantified by the well known entropy of entanglement however we show that mixed entangled states are partially ordered under locc even in the asymptotic limit therefore non uniqueness of entanglement measure is understood on the basis of an operational notion of asymptotic convertibility

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 the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata we consider the nonnegative inverse eigenvalue problem with partial eigendata which aims to find a nonnegative matrix such that it is nearest to a pre estimated nonnegative matrix and satisfies the prescribed eigendata in this paper we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem we also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries numerical tests including a practical engineering application in vibrations show the efficiency of the proposed iterative schemes

k β and k nearly uniformly convex banach spaces vol 162 no 2 1991 k and k nearly uniformly convex banach spaces denka kutzarova different uniform geometrical properties have been defined between the uniform convexity and the reflexivity of banach spaces in the present paper we introduce other properties of this type namely k and k nearly uniform convexity the definitions as well as some of the results presented here are announced in 7 sullivan 24 has defined k uniformly rotund banach spaces which by a suggestion of davis are all superreflexive fan and glicksberg 1 have introduced fully k convex banach spaces let x1 xk 1 be vectors in a banach space x the k dimensional volume enclosed by x1 xk 1 is given by v x1 x2 xk 1

on the capacities of bipartite hamiltonians and unitary gates we consider interactions as bidirectional channels we investigate the capacities for interaction hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information we give analytic expressions for the entanglement generating capacity and entanglement assisted one way classical communication capacity of interactions and show that these quantities are additive so that the asymptotic capacities equal the corresponding 1 shot capacities we give general bounds on other capacities discuss some examples and conclude with some open questions

simplifying additivity problems using direct sum constructions we study the additivity problems for the classical capacity of quantum channels the minimal output entropy and its convex closure we show for each of them that additivity for arbitrary pairs of channels holds if and only if it holds for arbitrary equal pairs which in turn can be taken to be unital in a similar sense weak additivity is shown to imply strong additivity for any convex entanglement monotone the implications are obtained by considering direct sums of channels or states for which we show how to obtain several information theoretic quantities from their values on the summands this provides a simple and general tool for lifting additivity results

an inverse eigenvalue problem for a vibrating string with two dirichlet spectra a classical inverse problem is can you hear the density of a string clamped at both ends the mathematical model gives rise to an inverse sturm liouville problem for the unknown density rho and it is well known that the answer is negative the dirichlet spectrum from the clamped end point conditions is insufficient there are many known ways to add additional information to gain a positive answer and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode the so called norming constants we make the assumption that neither of these changes are possible instead we will add known mass densities to the string in a way we can prescribe and remeasure the dirichlet spectrum we will not be able to answer the uniqueness question in its most general form but will give some insight to what added masses should be chosen and how this can lead to a reconstruction of the original string density

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

the conditional common information in classical and quantum secret key distillation in this paper we consider two extensions of the gacs korner common information to three variables the conditional common information cci and the coarse grained conditional common information ccci both quantities are shown to be useful technical tools in the study of classical and quantum resource transformations in particular the ccci is shown to have an operational interpretation as the optimal rate of secret key extraction from an eavesdropped classical source p xyz when alice x and bob y are unable to communicate but share common randomness with the eavesdropper eve z moving to the quantum setting we consider two different ways of generating a tripartite quantum state from classical correlations p xyz 1 coherent encodings sum xyz sqrt p xyz xyz rangle and 2 incoherent encodings sum xyz p xyz xyz rangle langle xyz we study how well can alice and bob extract secret key from these quantum sources using quantum operations compared with the extraction of key from the underlying classical sources p xyz using classical operations while the power of quantum mechanics increases alice and bob s ability to generate shared randomness it also equips eve with a greater arsenal of eavesdropping attacks therefore it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one we first demonstrate that the classical key rate of p xyz is equivalent to the quantum key rate for an incoherent quantum encoding of the distribution for coherent encodings we next show that the classical and quantum rates are generally incomparable and in fact their difference can be arbitrarily large in either direction finally we introduce a zoo of entangled tripartite states all characterized by the conditional common information of their encoded probability distributions remarkably for these states almost all entanglement measures such as alice and bob s entanglement cost squashed entanglement and relative entropy of entanglement can be sharply bounded or even exactly expressed in terms of the conditional common information in the latter case we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated

general theory of environment assisted entanglement distillation we evaluate the one shot entanglement of assistance for an arbitrary bipartite state this yields another interesting result namely a characterization of the one shot distillable entanglement of a bipartite pure state this result is shown to be stronger than that obtained by specializing the one shot hashing bound to pure states finally we show how the one shot result yields the operational interpretation of the asymptotic entanglement of assistance proved by smolin and coworkers

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

random bures mixed states and the distribution of their purity ensembles of random density matrices determined by various probability measures are analysed a simple and efficient algorithm to generate at random density matrices distributed according to the bures measure is proposed this procedure may serve as an initial step in performing the bayesian approach to quantum state estimation based on the bures prior we study the distribution of purity of random mixed states the moments of the distribution of purity are determined for quantum states generated with respect to the bures measure this calculation serves as an exemplary application of the deform and study approach in the theory of integrable systems leading to one of painleves transcendents

inverse problems for r s symmetric matrices in structural dynamic model updating let r s c n n be nontrivial involutions i e r r 1 i n and s s 1 i n a matrix a c n n is referred to as r s symmetric if and only if r a s a the set of all r s symmetric matrices of order n is denoted by c s n n r s given a full column rank matrix x c n m a matrix b c m m and a matrix a c n n in structural dynamic model updating we usually consider the sets s 1 a a c s n n r s x h a x b and s 2 a a c s n n r s x h a x b min in the frobenius norm sense where the superscript h denotes conjugate transpose then we characterize the unique matrices a arg min a s 1 a a and a arg min a s 2 a a when r r h and s s h by using the generalized singular value decomposition gsvd the necessary and sufficient conditions for the non emptiness of s 1 and the general representations of the elements in s 1 and a are derived respectively the analytical expressions of a s 2 and a are also obtained by using the gsvd the canonical correlation decomposition ccd and the projection theorem finally a corresponding numerical algorithm and some illustrated examples are presented

the zero error entanglement cost is highly non additive the schmidt number is an entanglement measure whose logarithm quantifies the zero error entanglement cost of generating a given quantum state using local operations and classical communication in this paper we show that the schmidt number is highly nonmultiplicative in the sense that for any integer n there exist states whose schmidt number remains constant when taking n copies of the given state these states also provide a rare instance in which the regularized zero error entanglement cost can be computed exactly we then explore the question of increasing the schmidt number by quantum operations we describe a class of bipartite quantum operations that preserve the schmidt number for pure state transformations and yet they can increase the schmidt number by an arbitrarily large amount when generating mixed states our results are obtained by making connections to the resource theory of quantum coherence and generalizing the class of dephasing covariant incoherent operations to the bipartite setting

revised geometric measure of entanglement we present a revised geometric measure of entanglement rgme which is just certain generalization of geometric measure of entanglement gme the revised version is an entanglement monotone some useful inequalities about rgme are deduced for example we give the formulae of rgme for the two parameter class of states in a 2 n quantum system the two particles high dimensional maximally entangled mixed state the isotropic state including the n particle d level case and two multipartite bound entangled states the results show there is a relation and then rgme is an appropriate measure of entanglement

on a class of free levy laws related to a regression problem abstract the free meixner laws arise as the distributions of orthogonal polynomials with constant coefficient recursions we show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum we apply this result to describe free levy processes with quadratic conditional variances and to prove a converse implication related to asymptotic freeness of random wishart matrices

composed ensembles of random unitary matrices composed ensembles of random unitary matrices are defined via products of matrices each pertaining to a given canonical circular ensemble of dyson we investigate statistical properties of spectra of some composed ensembles and demonstrate their physical relevance we also discuss the methods of generating random matrices distributed according to invariant haar measure on the orthogonal and unitary group

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

an entanglement measure based on two order minors in this paper a new entanglement measure called emm the entanglement measure based on minors has been constructed by the convex roof method and proved to be a good entanglement measure according to the axiomatic point of view computation of emm can be finished directly by the two order minors of a coefficient matrix instead of the eigenvalues of the density operator required by most of the other entanglement measures so it is very fast and very easy on the other hand emm is related to the modified bures distance between and the closest separable state so it has geometry meaning we also investigate the relations between emm and the entanglement of formation negativity and logarithmic negativity and discover that emm is always smaller than or equal to them emm is equivalent to concurrence however their definitions and methods of proof are completely different emm will improve the efficiency of searching for maximally entangled multipartite states

subnormalized states and trace nonincreasing maps we investigate the set of completely positive trace nonincreasing linear maps acting on the set mn of mixed quantum states of size n extremal point of this set of maps are characterized and its volume with respect to the hilbert schmidt hs euclidean measure is computed explicitly for an arbitrary n the spectra of partially reduced rescaled dynamical matrices associated with trace nonincreasing completely positive maps belong to the n cube inscribed in the set of subnormalized states of size n as a by product we derive the measure in mn induced by partial trace of mixed quantum states distributed uniformly with respect to the hs measure in mn2

generating random density matrices we study various methods to generate ensembles of random density matrices of a fixed size n obtained by partial trace of pure states on composite systems structured ensembles of random pure states invariant with respect to local unitary transformations are introduced to analyze statistical properties of quantum entanglement in bi partite systems we analyze the distribution of schmidt coefficients of random pure states such a distribution is derived in the case of a superposition of k random maximally entangled states for another ensemble obtained by performing selective measurements in a maximally entangled basis on a multi partite system we show that this distribution is given by the fuss catalan law and find the average entanglement entropy a more general class of structured ensembles proposed containing also the case of bures forms an extension of the standard ensemble of structureless random pure states described asymptotically as n by the marchenko pastur distribution

the asymptotic entanglement cost of preparing a quantum state we give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state is equal to limn ef n n where ef is the entanglement of formation

geometries of quantum states the quantum analog of the fisher information metric of a probability simplex is searched and several riemannian metrics on the set of positive definite density matrices are studied some of them appeared in the literature in connection with cramer rao type inequalities or the generalization of the berry phase to mixed states they are shown to be stochastically monotone here all stochastically monotone riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them a class of metrics can be extended to pure states and a constant multiple of the fubini study metric appears in the extension

bures volume of the set of mixed quantum states we compute the volume of the n2 1 dimensional set n of density matrices of size n with respect to the bures measure and show that it is equal to that of an n2 1 dimensional hyper hemisphere of radius 1 2 for n 2 we obtain the volume of the uhlmann hemisphere s3 4 we find also the area of the boundary of the set n and obtain analogous results for the smaller set of all real density matrices an explicit formula for the bures hall normalization constants is derived for an arbitrary n

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

cryptographic distinguishability measures for quantum mechanical states this paper mostly expository in nature surveys four measures of distinguishability for quantum mechanical states this is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions namely the probability of an identification error the kolmogorov distance the bhattacharyya coefficient and the shannon 1948 distinguishability as defined through mutual information these measures have a long history of use in statistical pattern recognition and classical cryptography we obtain several inequalities that relate the quantum distinguishability measures to each other one of which may be crucial for proving the security of quantum cryptographic key distribution in another vein these measures and their connecting inequalities are used to define a single notion of cryptographic exponential indistinguishability for two families of quantum states this is a tool that may prove useful in the analysis of various quantum cryptographic protocols

quantum correlations and secret bits it is shown that i all entangled states can be mapped by single copy measurements into probability distributions containing secret correlations and ii if a probability distribution obtained from a quantum state contains secret correlations then this state has to be entangled these results prove the existence of a two way connection between secret and quantum correlations in the process of preparation they also imply that either it is possible to map any bound entangled state into a distillable probability distribution or bipartite bound information exists

quantum state filter with disturbance and noise a quantum state filter qsf is proposed in this paper to estimate a low rank quantum density matrix from informationally incomplete and contaminated measurements there exist sparse disturbances on the quantum density matrix and gaussian noise in the measurements a proximal jacobian variant of the alternating direction method of multipliers pj admm is proposed to design the qsf the closed form solutions to three resulting subproblems are given and the iterative qsf is developed the proposed qsf is proved to be convergent and its superiority is demonstrated in the numerical illustrations compared with different state of the art methods

product of ginibre matrices fuss catalan and raney distributions squared singular values of a product of s square random ginibre matrices are asymptotically characterized by probability distributions p s x such that their moments are equal to the fuss catalan numbers of order s we find a representation of the fuss catalan distributions p s x in terms of a combination of s hypergeometric functions of the type s f s 1 the explicit formula derived here is exact for an arbitrary positive integer s and for s 1 it reduces to the marchenko pastur distribution using similar techniques involving the mellin transform and the meijer g function we find exact expressions for the raney probability distributions the moments of which are given by a two parameter generalization of the fuss catalan numbers these distributions can also be considered as a two parameter generalization of the wigner semicircle law

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

on the schmidt robustness of pure states robustness measures as introduced by vidal and tarrach 1999 robustness of entanglement phys rev a 59 141 55 quantify the extent to which entangled states remain entangled under mixing analogously we introduce here the schmidt robustness and the random schmidt robustness the latter notion is closely related to the construction of schmidt balls around the identity we analyse the situation for pure states and provide nontrivial upper and lower bounds upper bounds to the random schmidt 2 robustness allow us to construct a particularly simple distillability criterion we present two conjectures the first one is related to the radius of inner balls around the identity in the convex set of schmidt number n states we also conjecture a class of optimal schmidt witnesses for pure states

compactly uniformly convex spaces and property β of rolewicz abstract we study different facets of property of rolewicz we remark that the notions of compact uniform convexity and property are isometrically equivalent and present new examples of spaces with that property an observation is made that the property can be formulated in terms of graphs and an estimate of the modulus is also given

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

logarithmic negativity a full entanglement monotone that is not convex it is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation a set of operations that incorporate local operations and classical communication as a subset and in the process a further proof is provided that the negativity does not increase on average under the same set of operations given that the logarithmic negativity is not a convex function this result is surprising as it is generally considered that convexity describes the local physical process of losing information the role of convexity and in particular its relation or lack thereof to physical processes is discussed and importance of continuity in this context is stressed

szlenk indices and uniform homeomorphisms we prove some rather precise renorming theorems for banach spaces with szlenk index wo we use these theorems to show the invariance of certain quantitative szlenk type indices under uniform homeomorphisms

dirichlet neumann inverse spectral problem for a star graph of stieltjes strings we solve two inverse spectral problems for star graphs of stieltjes strings with dirichlet and neumann boundary conditions respectively at a selected vertex called root the root is either the central vertex or in the more challenging problem a pendant vertex of the star graph at all other pendant vertices dirichlet conditions are imposed at the central vertex at which a mass may be placed continuity and kirchhoff conditions are assumed we derive conditions on two sets of real numbers to be the spectra of the above dirichlet and neumann problems our solution for the inverse problems is constructive we establish algorithms to recover the mass distribution on the star graph i e the point masses and lengths of subintervals between them from these two spectra and from the lengths of the separate strings if the root is a pendant vertex the two spectra uniquely determine the parameters on the main string i e the string incident to the root if the length of the main string is known the mass distribution on the other edges need not be unique the reason for this is the non uniqueness caused by the non strict interlacing of the given data in the case when the root is the central vertex finally we relate of our results to tree patterned matrix inverse problems

the uniqueness theorem for entanglement measures we explore and develop the mathematics of the theory of entanglement measures after a careful review and analysis of definitions of preliminary results and of connections between conditions on entanglement measures we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von neumann entropy on pure states we also prove several versions of a theorem on extreme entanglement measures in the case of mixed states we analyze properties of the asymptotic regularization of entanglement measures proving for example convexity for the entanglement cost and for the regularized relative entropy of entangle ment

spectral density of generalized wishart matrices and free multiplicative convolution we investigate the level density for several ensembles of positive random matrices of a wishart like structure w xx where x stands for a non hermitian random matrix in particular making use of the cauchy transform we study the free multiplicative powers of the marchenko pastur mp distribution mp s which for an integer s yield fuss catalan distributions corresponding to a product of s independent square random matrices x x 1 x s new formulas for the level densities are derived for s 3 and s 1 3 moreover the level density corresponding to the generalized bures distribution given by the free convolution of arcsine and mp distributions is obtained we also explain the reason of such a curious convolution the technique proposed here allows for the derivation of the level densities for several other cases

random graph states maximal flow and fuss catalan distributions for any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems each edge of the graph represents a bipartite maximally entangled state each vertex represents a random unitary matrix generated according to the haar measure which describes the coupling between subsystems dividing all subsystems into two parts one may study entanglement with respect to this partition a general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented our technique relies on weingarten calculus and flow problems we analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free poissonian marchenko pastur distribution we derive a discrete family of generalized fuss catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

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