quantum key distribution based on private states unconditional security over untrusted channels with zero quantum capacity in this paper we prove unconditional security for a quantum key distribution qkd protocol based on distilling pbits twisted ebits from an arbitrary untrusted state that is claimed to contain distillable key our main result is that we can verify security using only public communication via parameter estimation of the given untrusted state the technique applies even to bound entangled states thus extending qkd to the regime where the available quantum channel has zero quantum capacity we also show how to convert our purification based qkd schemes to prepare measure schemes

on the relationship between measures of discrimination and the performance of suboptimal detectors the problem of designing and analyzing suboptimal detectors via statistical distance measures is considered as a preliminary result we show that only the minimum and maximum probability of error are valid measures of discrimination between the input statistics this result would seem then to imply that the use of distance measures in this context can be inappropriate however to overcome this apparent obstacle we demonstrate explicit relationships between various f divergences and the loss in performance of an arbitrary detector relative to the optimal detector in particular we establish both upper and lower bounds on the performance loss of a suboptimal detector in terms of the distance between the pertinent statistics of both the optimal and suboptimal detectors while designing detectors by minimizing these upper bounds can be an elusive task in many practical cases the lower bound presented herein holds with equality in this case minimizing the separation of the output statistics of the detector with respect to a particular f divergence equivalently minimizes the resulting probability of error of the detector to facilitate design other researchers have established conditions under which one may design arbitrary detection strategies with respect to a specific f divergence kullback leibler distance being a principal example we extend this approach by deriving necessary and sufficient conditions under which one may design detection strategies with respect to an arbitrarily chosen f divergence thus when these conditions are met one may optimize a detector with respect to the most analytically tractable distance measure to obtain the minimum probability of error detector over a selected class of detection strategies examples demonstrating the utility of this theory for the problem of designing optimal linear detectors and optimal signal sets are presented

dissecting the qutrit to visualize a higher dimensional object it is convenient to consider its two dimensional cross sections the set of quantum states for a three level system has eight dimensions we supplement a recent paper by goyal et al arxiv 1111 4427 by considering the set of all possible two dimensional cross sections of the qutrit each such cross section is bounded by a plane cubic curve

flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices abstract let t be an n n matrix the davis wielandt shell of t is defined as the set w t t t x tx x t tx x c r x x 1 in this paper we discuss the existence of flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices

numerical range for random matrices abstract we analyze the numerical range of high dimensional random matrices obtaining limit results and corresponding quantitative estimates in the non limit case for a large class of random matrices their numerical range is shown to converge to a disc in particular numerical range of complex ginibre matrix almost surely converges to the disk of radius 2 since the spectrum of non hermitian random matrices from the ginibre ensemble lives asymptotically in a neighborhood of the unit disk it follows that the outer belt of width 2 1 containing no eigenvalues can be seen as a quantification the non normality of the complex ginibre random matrix we also show that the numerical range of upper triangular gaussian matrices converges to the same disk of radius 2 while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to 2 e

discriminant of symmetric matrices as a sum of squares and the orthogonal group it is proved that the discriminant of n n real symmetric matrices can be written as a sum of squares where the number of summands equals the dimension of the space of n variable spherical harmonics of degree n the representation theory of the orthogonal group is applied to express the discriminant of 3 3 real symmetric matrices as a sum of five squares and to show that it cannot be written as the sum of less than five squares it is proved that the discriminant of 4 4 real symmetric matrices can be written as a sum of seven squares these improve results of kummer from 1843 and borchardt from 1846 2010 wiley periodicals inc

real numerical shadow and generalized b splines abstract restricted numerical shadow p a x z of an operator a of order n is a probability distribution supported on the numerical range w x a restricted to a certain subset x of the set of all pure states normalized one dimensional vectors in c n its value at point z c equals the probability that the inner product u a u is equal to z where u stands for a random complex vector from the set x distributed according to the natural measure on this set induced by the unitarily invariant fubini study measure for a hermitian operator a of order n we derive an explicit formula for its shadow restricted to real states p a r x show relation of this density to the dirichlet distribution and demonstrate that it forms a generalization of the b spline furthermore for operators acting on a space with tensor product structure h a h b we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order n 4

minimal and maximal operator spaces and operator systems in entanglement theory abstract we examine k minimal and k maximal operator spaces and operator systems and investigate their relationships with the separability problem in quantum information theory we show that the matrix norms that define the k minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k positive linear maps and bound entanglement similarly we investigate the k super minimal and k super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k block positive operators and unnormalized states with schmidt number no greater than k respectively we characterize a class of norms on the k super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the schmidt number of a quantum state that generalizes the recently developed separability criterion based on trace contractive maps

detecting quantum entanglement we review the criteria for separability and quantum entanglement both in a bipartite as well as a multipartite setting we discuss bell inequalities entanglement witnesses entropic inequalities bound entanglement and several features of multipartite entanglement we indicate how these criteria bear on the experimental detection of quantum entanglement

general paradigm for distilling classical key from quantum states in this paper we develop a formalism for distilling a classical key from a quantum state in a systematic way expanding on our previous work on a secure key from bound entanglement horodecki et al 2005 more detailed proofs discussion and examples are provided of the main results namely we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory with the only change being that instead of distilling einstein podolsky rosen epr pairs the parties distill private states the form of these general private states are given and we show that there are a number of useful ways of expressing them some of the private states can be approximated by certain states which are bound entangled thus distillable entanglement is not a requirement for a private key we find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel pqc we also find a general class of states which have negative partial transpose are npt but which appear to be bound entangled the relative entropy distance is shown to be an upper bound on the rate of a key this allows us to compute the exact value of a distillable key for a certain class of private states

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

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

distillability via protocols respecting the positivity of partial transpose we show that all quantum states that do not have a positive partial transpose are distillable via channels which preserve the positivity of the partial transpose the question whether bound entangled states with non positive partial transpose exist is therefore closely related to the connection between the set of separable superoperators and positive partial transpose preserving maps

low dimensional quite noisy bound entanglement with a cryptographic key we provide a class of bound entangled states that have a positive distillable secure key rate the smallest state of this kind is 4 4 our class is a generalization of the class presented in horodecki et al 2008 ieee trans inf theory 54 2621 5 it is much wider containing in particular states from the boundary of ppt entangled states all of the states in the previous class were of this kind and also states inside the set of ppt entangled states even approaching the separable states this generalization comes at a price for the wider class a positive key rate requires in general apart from the one way devetak winter protocol used in the previous case also the recurrence preprocessing and thus is effectively a two way protocol we also analyze the amount of noise that can be admixtured to the states of our class without losing the key distillability property which may be crucial for experimental realization the wider class contains key distillable states with higher entropy up to 3 524 as opposed to 2 564 for the previous class

a few steps more towards npt bound entanglement in this paper existence of bound entangled states with nonpositive partial transpose npt is considered as one knows existence of such states would in particular imply nonadditivity of distillable entanglement moreover it would rule out a simple mathematical description of the set of distillable states the particular state known to be 1 copy nondistillable and supposed to be bound entangled is considered the problem of its two copy distillability which boils down to show that maximal overlap of some projector q with schmidt rank two states does not exceed 1 2 called the half property is studied first it is shown that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies then the problem in attacked twofold way a the half property is proved for some wide classes of schmidt rank two states b the overlap for all schmidt rank two states is bounded from above by c 3 4 moreover the problem has been translated into the following matrix analysis problem bound the sum of the squares of the two largest singular values of matrix a i i b with a b traceless 4 4 matrices and tr afa tr bfb 1 4

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

boundary generating curves of the c numerical range abstract let a be an n n matrix and c c 1 c 2 c n be a real n tuple the c numerical range of a is the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis of c n we obtain parametric representations of the boundary generating curve of the c numerical range of a matrix applying this result we generalize the result of anderson to the c numerical range furthermore we give a description of the boundary generating curve of the c numerical range of certain types of nilpotent toeplitz matrices a sufficient condition for the boundary generating curve to be rational is obtained finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

joint numerical range and its generating hypersurface in this paper we study the joint numerical range of m tuples of hermitian matrices via their generating hypersurfaces an example is presented which shows the invalidity of an analogous kippenhahn theorem for the joint numerical range of three hermitian matrices

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

continuity of the maximum entropy inference convex geometry and numerical ranges approach we study the continuity of an abstract generalization of the maximum entropy inference a maximizer it is defined as a right inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body using convex geometry we prove amongst others the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations further we use numerical range methods in the case of quantum inference which refers to two observables one result is a complete characterization of points of discontinuity for 3 3 matrices

correlated entanglement distillation and the structure of the set of undistillable states we consider entanglement distillation under the assumption that the input states are allowed to be correlated among each other we hence replace the usually considered independent and identically distributed hypothesis by the weaker assumption of merely having identical reductions we find that whether a state is then distillable or not is only a property of these reductions and not of the correlations that are present in the input state this is shown by establishing an appealing relation between the set of copy correlated undistillable states and the standard set of undistillable states the former turns out to be the convex hull of the latter as an example of the usefulness of our approach to the study of entanglement distillation we prove a new activation result which generalizes earlier findings it is shown that for every entangled state and every k there exists a copy correlated k undistillable state such that is single copy distillable finally the relation of our results to the conjecture about the existence of bound entangled states with a nonpositive partial transpose is discussed

joint numerical ranges quantum maps and joint numerical shadows abstract we associate with a k tuple of hermitian n n matrices a probability measure on r k supported on their joint numerical range the joint numerical shadow of these matrices when k 2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices we apply this material to the theory of quantum information thus we show that quantum maps on the set of quantum states defined by kraus operators satisfying the identity resolution assumption shrink joint numerical ranges

an algorithm to explore entanglement in small systems a quantum state s entanglement across a bipartite cut can be quantified with entanglement entropy or more generally schmidt norms using only schmidt decompositions we present a simple iterative algorithm to maximize schmidt norms depending on the choice of norm the optimizing states maximize or minimize entanglement possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace recognizing that convergence but not success is certain we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels

complementation in the face lattice of a proper cone abstract barker proved that the lattice of faces f k of a finite dimensional proper cone k is always complemented the proof however contained a gap in this paper we offer a correct proof of the result at the same time new characterizations of lattices of finite length which are section complemented or relatively complemented are found it is also proved that the lattice of exposed faces of a proper cone k is complemented finally we show that the class of proper cones with a relatively complemented face lattice is strictly contained in the class of proper cones with a dual atomistic face lattice for a few natural related questions counterexamples are given it turns out that one of our examples also answers in the negative an open problem posed recently by the second author

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

low dimensional bound entanglement with one way distillable cryptographic key in this paper we provide a class of bound entangled states that have positive distillable secure key rate the smallest state of this kind is 4 otimes 4 which shows that peculiar security contained in bound entangled states does not need high dimensional systems we show that for these states a positive key rate can be obtained by one way devetak winter dw protocol subsequently the volume of bound entangled key distillable states for m otimes n hilbert space with m n 4 is shown to be nonzero we provide a scheme of verification of cryptographic quality of experimentally prepared state in terms of local observables proposed set of seven collective settings is proven to be optimal in number of settings

3 3 matrices with a flat portion on the boundary of the numerical range the set of unitarily irreducible complex 3 3 matrices whose numerical range has a flat portion is explicitly described it is proved that the set is connected and real analytic properties of the set are studied connectedness of the sets of unitarily irreducible 3 3 matrices having two other possible shapes of their numerical ranges is proved as well

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

joint ranges of hermitian matrices and simultaneous diagonalization abstract let a a 1 a k be a k tuple of heritian operators on an n diensional inner product space x with unit sphere s u x u u 1 with 1 k as the corresponding quadratic forms i e j x x a j x for all x x j 1 k we study ranges of over various subsets t of x the interplay between geometric properties of t and algebraic properties of a is our main concern in particular we characterize those a for which s is identical with the convex hull of the joint spectrum of a defined by a c k a j u j u for some u s the closely related problem of diagonalizing several hermitian operators simultaneously is also studied

convexity of the joint numerical range topological and differential geometric viewpoints abstract we investigate the convexity of the joint numerical range of m tuples of n n hermitian matrices the methods come from differential geometry and the differential and algebraic topology our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n modulo a mild technical assumption this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the m 1 dimensional unit sphere the condition is that the eigenvalue has constant multiplicity we show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges as a byproduct of our main result we obtain a new proof of the celebrated toeplitz hausdorff theorem and of its extension the numerical range of any triple of hermitian n n matrices is convex if n 2

sets of matrices with given joint numerical range abstract the following question is addressed to what extent the n tuple of m m hermitian matrices is determined by its joint numerical range the cases m 2 n arbitrary and m n 3 are considered in detail

distillability and ppt entanglement of low rank quantum states the bipartite quantum states with rank strictly smaller than the maximum of the ranks of the reduced states a and b are distillable by local operations and classical communication horodecki p smolin j a terhal b m and thapliyal a v 2003 theor comput sci 292 589 96 1999 arxiv quant ph 9910122 our first main result is that this is also true for npt states with rank equal to this maximum a state is ppt if the partial transpose of its density matrix is positive semidefinite and otherwise it is npt this was conjectured first in 1999 in the special case when the ranks of a and b are equal see horodecki p smolin j a terhal b m and thapliyal a v 2003 theor comput sci 292 589 96 1999 arxiv quant ph 9910122 our second main result provides a complete solution of the separability problem for bipartite states of rank 4 namely we show that such a state is separable if and only if it is ppt and its range contains at least one product state we also prove that the so called checkerboard states are distillable if and only if they are npt