eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

construction of multi qubit optimal genuine entanglement witnesses we interpret multi partite genuine entanglement witnesses as simultaneous positivity of various maps arising from them we apply this result to multi qubit x shaped hermitian matrices and characterize the conditions for them to be genuine entanglement witnesses in terms of entries furthermore we find all optimal entries among them they turn out to have spanning properties and so they detect a non zero volume set of multi qubit genuine entanglement we also characterize decomposability for x shaped entanglement witnesses

entangled three qubit states without concurrence and three tangle we provide a complete analysis of mixed three qubit states composed of a greenberger horne zeilinger state and a w state orthogonal to the former we present optimal decompositions and convex roofs for the three tangle further we provide an analytical method to decide whether or not an arbitrary rank 2 state of three qubits has vanishing three tangle these results highlight intriguing differences compared to the properties of two qubit mixed states and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states by studying the coffman kundu wootters inequality we find that while the amounts of inequivalent entanglement types strictly add up for pure states this monogamy can be lifted for mixed states by virtue of vanishing tangle measures

separable states with unique decompositions we search for faces of the convex set consisting of all separable states which are affinely isomorphic to simplices to get separable states with unique decompositions in the two qutrit case we found that six product vectors spanning a five dimensional space give rise to a face isomorphic to the 5 dimensional simplex with six vertices under a suitable linear independence assumption if the partial conjugates of six product vectors also span a 5 dimensional space then this face is inscribed in the face for ppt states whose boundary shares the fifteen 3 simplices on the boundary of the 5 simplex the remaining boundary points consist of ppt entangled edge states of rank four we also show that every edge state of rank four arises in this way if the partial conjugates of the above six product vectors span a 6 dimensional space then we have a face isomorphic to 5 simplex whose interior consists of separable states with unique decompositions but with non symmetric ranks we also construct a face isomorphic to the 9 simplex as applications we give answers to questions in the literature chen and djokovic j math phys 54 022201 2013 and chen and djokovic commun math phys 323 241 284 2013 and construct 3 3ppt states of type 9 5 for the qubit qudit cases with d 3 we also show that d 1 dimensional subspaces give rise to faces isomorphic to the d simplices in most cases

optimal indecomposable witnesses without extremality or the spanning property one of the interesting problems of optimal indecomposable entanglement witnesses is whether there exists an optimal indecomposable entanglement witness which neither has the spanning property nor is associated with an extremal positive linear map here we answer this question in the negative by examining the extremality of the positive linear maps constructed by qi and hou 2011 j phys a math theor 44 215305

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

images in quantum entanglement a system for classifying and quantifying entanglement in spin 1 2 pure states is presented based on simple images from the image point of view an entangled state can be described as a linear superposition of separable object wavefunction o plus a portion of its own inverse image bell states can be defined in this way using the method of images the three spin 1 2 system is discussed in some detail this system can exhibit exclusive three particle 123 entanglement two particle entanglements 12 13 23 and or mixtures of all four all four image states are orthogonal both to each other and to the object wavefunction in general five entanglement parameters 12 13 23 123 and 123 are required to define the general entangled state in addition it is shown that there is considerable scope for encoding numbers at least from the classical point of view but using quantum mechanical principles methods are developed for their extraction it is shown that concurrence can be used to extract even partite but not odd partite information additional relationships are also presented which can be helpful in the decoding process however in general numerical methods are mandatory a simple roulette method for decoding is presented and discussed but it is shown that if the encoder chooses to use transcendental numbers for the angles defining the target function 1 1 etc the method rapidly turns into the devil s roulette requiring finer and finer angular steps

on π quasinormally embedded subgroups of finite group a subgroup of a group g is said to be quasinormal in g if it permutes with every sylow subgroup of g a subgroup h of a group g is said to be quasinormally embedded in g if for each prime dividing the order of h a sylow p subgroup of h is also a sylow p subgroup of some quasinormal subgroups of g in this paper we investigate the influence of quasinormally embedded of maximal or minimal subgroup of sylow subgroups of the generalized fitting subgroup of a finite group the main theorems are as follows theorem 1 1 let f be a saturated formation containing u suppose that g is a group then g f if and only if there is a normal subgroup h of g such that g h f and all maximal subgroups of any sylow subgroup of f h are quasinormally embedded in g theorem 1 2 let f be a saturated formation containing u and let g be a group then g f if and only if there is a normal subgroup h in g such that g h f and the subgroups of prime order or order 4 of the generalized fitting subgroup f h are quasinormally embedded in g theorem 1 3 let f be a saturated formation such that n f let g be a group such that every element of order 4 of the generalized fitting subgroup f gf is quasinormally embedded in g then g belongs to f if and only if x lies in the f hypercenter zf g of g for every element x of prime order of f gf these results extended recent results of asaad et al on s quasinormally embedded subgroups of finite groups j pure appl algebra 165 2001 129 135 li et al the influence of minimal subgroups on the structure of finite groups proc amer math soc in press the influence of quasinormality of maximal subgroups of sylow subgroups of a finite group arch math in press

compressive sensing on manifolds using a nonparametric mixture of factor analyzers algorithm and performance bounds nonparametric bayesian methods are employed to constitute a mixture of low rank gaussians for data x rn that are of high dimension n but are constrained to reside in a low dimensional subregion of rn the number of mixture components and their rank are inferred automatically from the data the resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds based on compressive sensing cs projection measurements the statistical cs inversion is performed analytically we derive the required number of cs random measurements needed for successful reconstruction based on easily computed quantities drawing on block sparsity properties the proposed methodology is validated on several synthetic and real datasets

a new criterion for indecomposability of positive maps we give a criterion for a positive mapping on the space of operators on a hilbert space to be indecomposable we show that this criterion can be applied to two families of positive maps these families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states

classification of mixed three qubit states we introduce a classification of mixed three qubit states in which we define the classes of separable biseparable w and greenberger horne zeilinger states these classes are successively embedded into each other we show that contrary to pure w type states the mixed w class is not of measure zero we construct witness operators that detect the class of a mixed state we discuss the conjecture that all entangled states with positive partial transpose pptes belong to the w class finally we present a new family of pptes edge states with maximal ranks

construction of 3 3 entangled edge states with positive partial transposes we construct a class of 3 3 entangled edge states with positive partial transposes using indecomposable positive linear maps this class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes

spectral properties of entanglement witnesses entanglement witnesses are observables which when measured detect entanglement in a measured composed system it is shown what kind of relations between eigenvectors of an observable should be fulfilled to allow an observable to be an entanglement witness some restrictions on the signature of entanglement witnesses based on an algebraic geometrical theorem will be given the set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive but not completely positive a translation of the results to the language of positive maps is also given the properties of entanglement witnesses and positive maps are expressed as special cases of general theorems for k schmidt witnesses and k positive maps the results are therefore presented in a general framework

a new criterion and a special class of k positive maps abstract we study k positive maps on operators we obtain a new criterion on k positivity in terms of the k numerical range and use it to improve and refine some earlier results on k positive maps related to the study of quantum information science we also consider a special class of positive maps extending the construction of choi on positive maps that are not completely positive some open questions on the decomposability of such maps are answered

generalized choi maps in three dimensional matrix algebra we consider a class of positive linear maps in the three dimensional matrix algebra which are generalizations of the positive linear map constructed by choi in the relation with positive semidefinite biquadratic forms we find conditions for which such maps are completely positive completely copositive decomposable and two positive

density of spherically embedded stiefel and grassmann codes the density of a code is the fraction of the coding space covered by packing balls centered around the codewords a high density indicates that a code performs well when used as a uniform point wise discretization of an ambient space this paper investigates the density of codes in the complex stiefel and grassmann manifolds equipped with the chordal distance arising from an euclidean embedding including the unitary group as a special case the choice of distance enables the treatment of the manifolds as subspaces of euclidean hyperspheres in this geometry the densest packings are not necessarily equivalent to maximum minimum distance codes computing a code s density follows from computing 1 the normalized volume of a metric ball and 2 the kissing radius the radius of the largest balls one can pack around the codewords without overlapping first the normalized volume of a metric ball is evaluated by asymptotic approximations the volume of a small ball can be well approximated by the volume of a locally equivalent tangential ball in order to properly normalize this approximation the precise volumes of the manifolds induced by their spherical embedding are computed for larger balls a hyperspherical cap approximation is used which is justified by a volume comparison theorem showing that the normalized volume of a ball in the stiefel or grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity then bounds on the kissing radius are derived alongside corresponding bounds on the density unlike spherical codes or codes in flat spaces the kissing radius of grassmann or stiefel codes cannot be exactly determined from its minimum distance it is nonetheless possible to derive bounds on density as functions of the minimum distance stiefel and grassmann codes have larger density than their image spherical codes when dimensions tend to infinity finally the bounds on density lead to refinements of the standard hamming bounds for stiefel and grassmann codes

grassmannian packings from operator reed muller codes this paper introduces multidimensional generalizations of binary reed muller codes where the codewords are projection operators and the corresponding subspaces are widely separated with respect to the chordal distance on grassmannian space parameters of these grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary reed muller codes the subspaces are associated with projection operators determined by pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest the grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas where separation with respect to the chordal distance on grassmannian space guarantees closeness to the channel capacity it is shown that the capacity of the noncoherent multiple input multiple output mimo channel at both low and moderate signal to noise ratio snr under the constraint that only isotropically distributed unitary matrices are used for information transmission is closely approximated by these packings

a class of generalized positive linear maps on matrix algebras abstract we construct a class of positive linear maps on matrix algebras we find conditions when these maps are atomic decomposable and completely positive we obtain a large class of atomic positive linear maps as applications in quantum information theory we discuss the structural physical approximation and optimality of entanglement witness associated with these maps

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

positive linear maps between matrix algebras which fix diagonals abstract we consider a class of positive linar maps from the n dimensional matrix algebra into itself which fix diagonal entries we show that they are expressed by hadamard products and study their decompositions into the sums of completely positive linear maps and completely copositive linear maps in the three dimensional case we show that every positive linear map in this type is decomposable and give an intrinsic characterization for the positivity of these maps when the involving coefficients are real numbers

systematic design of unitary space time constellations we propose a systematic method for creating constellations of unitary space time signals for multiple antenna communication links unitary space time signals which are orthonormal in time across the antennas have been shown to be well tailored to a rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients the signals can achieve low probability of error by exploiting multiple antenna diversity because the fading coefficients are not known the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum euclidean distance norm our construction begins with the first signal in the constellation an oblong complex valued matrix whose columns are orthonormal and systematically produces the remaining signals by successively rotating this signal in a high dimensional complex space this construction easily produces large constellations of high dimensional signals we demonstrate its efficacy through examples involving one two and three transmitter antennas

the role of phases in detecting three qubit entanglement we propose separability criteria for three qubit states in terms of diagonal and anti diagonal entries to detect entanglement with positive partial transposes we report that the phases that is the angular parts of anti diagonal entries play a crucial role in determining whether a given three qubit state is separable or entangled and they must obey even an identity for separability in some cases these criteria are strong enough to detect ppt positive partial transpose entanglement with nonzero volume in several cases when all the entries are zero except for diagonal and anti diagonal entries we characterize separability using phases these include the cases when anti diagonal entries of such states share a common magnitude and when ranks are less than or equal to six we also compute the lengths of rank six cases and find three qubit separable states with lengths 8 whose maximum ranks of partial transposes are 7

the structural physical approximation conjecture it was conjectured that the structural physical approximation spa of an optimal entanglement witness is separable or equivalently that the spa of an optimal positive map is entanglement breaking this conjecture was disproved first for indecomposable maps and more recently for decomposable maps the arguments in both cases are sketched along with important related results this review includes background material on topics including entanglement witnesses optimality duality of cones decomposability and the statement and motivation for the spa conjecture so that it should be accessible for a broad audience

characterization of partial positive transposition states and measures of entanglement a detailed characterization of partial positive transposition ppt states both in the heisenberg and in the schrodinger picture is given measures of entanglement are defined and discussed in details illustrative examples are provided

a complete set of local invariants for a family of multipartite mixed states we study the equivalence of quantum states under local unitary transformations by using the singular value decomposition a complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in composite systems two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices

the classification of multipartite quantum correlation in multipartite entanglement theory the partial separability properties have an elegant yet complicated structure which becomes simpler in the case when multipartite correlations are considered in this work we elaborate this by giving necessary and sufficient conditions for the existence and uniqueness of the class of a given class label by the use of which we work out the structure of the classification for some important particular cases namely for the finest classification for the classification based on k partitionability and k producibility and for the classification based on the atoms of the correlation properties

solving inverse problems with piecewise linear estimators from gaussian mixture models to structured sparsity a general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper the approach is based on gaussian mixture models which are estimated via a maximum a posteriori expectation maximization algorithm a dual mathematical interpretation of the proposed framework with a structured sparse estimation is described which shows that the resulting piecewise linear estimate stabilizes the estimation when compared with traditional sparse inverse problem techniques we demonstrate that in a number of image inverse problems including interpolation zooming and deblurring of narrow kernels the same simple and computationally efficient algorithm yields results in the same ballpark as that of the state of the art

extension of positive maps into b h soient 7b h et 7b k des espaces de hilbert complexes et soit a un sous espace lineaire auto adjoint norme clos d operateurs bornes b 7b k sur 7b k contenant l operateur identite si on note par b a 7b h les applications positives dans l ensemble b a 7b h des applications lineaires bornees de a dans b 7b h on etudie le probleme d extension pour des applications dans b a 7b h aux applications dans b b 7b k 7b h

petrels parallel subspace estimation and tracking by recursive least squares from partial observations many real world datasets exhibit an embedding of low dimensional structure in a high dimensional manifold examples include images videos and internet traffic data it is of great significance to estimate and track the low dimensional structure with small storage requirements and computational complexity when the data dimension is high therefore we consider the problem of reconstructing a data stream from a small subset of its entries where the data is assumed to lie in a low dimensional linear subspace possibly corrupted by noise we further consider tracking the change of the underlying subspace which can be applied to applications such as video denoising network monitoring and anomaly detection our setting can be viewed as a sequential low rank matrix completion problem in which the subspace is learned in an online fashion the proposed algorithm dubbed parallel estimation and tracking by recursive least squares petrels first identifies the underlying low dimensional subspace and then reconstructs the missing entries via least squares estimation if required subspace identification is performed via a recursive procedure for each row of the subspace matrix in parallel with discounting for previous observations numerical examples are provided for direction of arrival estimation and matrix completion comparing petrels with state of the art batch algorithms

positive maps from irreducibly covariant operators in this paper we discuss positive maps induced by irreducibly covariant linear operators for finite groups the application of group theory methods allows deriving some new results of a different kind in particular a family of necessary conditions for positivity for such objects is derived stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map in the low dimensional cases for the permutation group s 3 and the quaternion group q the necessary and sufficient conditions are given together with the discussion on their decomposability in higher dimensions we present positive maps induced by a three dimensional irreducible representation of the permutation group s 4 and d dimensional representation of the monomial unitary group mu d in the latter case we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction we show that the generalised choi map can be obtained by considered covariant maps induced by the monomial unitary group as an additional result a novel interpretation of the fujiwara algolet conditions for positivity and complete positivity is presented finally in the end a new form of an irreducible representation of the symmetric group s n is constructed allowing us to simplify the form of certain choi jamiolkowski images derived for irreducible representations of the symmetric group

the role of principal angles in subspace classification subspace models play an important role in a wide range of signal processing tasks and this paper explores how the pairwise geometry of subspaces influences the probability of misclassification when the mismatch between the signal and the model is vanishingly small the probability of misclassification is determined by the product of the sines of the principal angles between subspaces when the mismatch is more significant the probability of misclassification is determined by the sum of the squares of the sines of the principal angles reliability of classification is derived in terms of the distribution of signal energy across principal vectors larger principal angles lead to smaller classification error motivating a linear transform that optimizes principal angles the transform presented here trait preserves some specific characteristic of each individual class and this approach is shown to be complementary to a previously developed transform lrt that enlarges inter class distance while suppressing intraclass dispersion theoretical results are supported by demonstration of superior classification accuracy on synthetic and measured data even in the presence of significant model mismatch

criterion for slocc equivalence of multipartite quantum states we study the stochastic local operation and classical communication slocc equivalence for arbitrary dimensional multipartite quantum states for multipartite pure states we present a necessary and sufficient criterion in terms of their coefficient matrices this condition can be used to classify some slocc equivalent quantum states with coefficient matrices having the same rank for multipartite mixed state we provide a necessary and sufficient condition by means of the realignment of matrix some detailed examples are given to identify the slocc equivalence of multipartite quantum states

multi partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose we investigate conditions on a finite set of multi partite product vectors for which separable states with corresponding product states have unique decompositions and show that this is true in most cases if the number of product vectors is sufficiently small in the three qubit case generic five dimensional spaces give rise to faces of the convex set consisting of all separable states which are affinely isomorphic to the five dimensional simplex with six vertices we construct as a byproduct three qubit entangled positive partial transpose edge states of rank 4 with explicit formulas this covers those entanglements which cannot be constructed from an unextendible product basis

finite groups with generalized ore supplement conditions for primary subgroups abstract we associate with every group g a set g of subgroups of g with 1 g if h g then we say that h is a subgroup of g if g g for each epimorphism g g then we say that is a subgroup functor we say also that a subgroup functor is hereditary provided h e whenever h e g and h g regular provided for any group g whenever h g is a p group and n is a minimal normal subgroup of g then g n g h n is a power of p regular respectively quasiregular provided for any primitive group g whenever h g is a p group and n is a respectively abelian minimal normal subgroup of g then g n g h n is a power of p let k h be subgroups of g and a subgroup functor then we say that the pair k h satisfies the f supplement condition in g if g has a subgroup t such that h t g and h t k z f t h is f supplemented in g if for some subgroup s of g contained in h the pair s h satisfies the f supplement condition in g where g g h g and h h h g in this paper we study the structure of a group g under the condition that some primary subgroups of g are f supplemented in g in particular we prove the following result theorem a let f be a saturated formation containing the class u of all supersoluble groups e a normal subgroup of g with g e f x e or x f e and a regular or hereditary regular subgroup functor suppose that every subgroup of g contained in x is subnormally embedded in g if every maximal subgroup of every non cyclic sylow subgroup of x is u supplemented in g then g f moreover in the case when is regular then every chief factor of g below e is cyclic the results in this paper not only cover and unify a long list of some known results but also cause a wide series of new results

choi matrices norms and entanglement associated with positive maps on matrix algebras abstract we study positive maps of b k into b h for finite dimensional hilbert spaces k and h our main emphasis is on how choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps special attention will be given to entanglement properties and k positive maps

merging of positive maps a construction of various classes of positive maps on matrix algebras abstract for two positive maps i b k i b h i i 1 2 we construct a new linear map b h b k where k k 1 k 2 c h h 1 h 2 c by means of some additional ingredients such as operators and functionals we call it a merging of maps 1 and 2 the properties of this construction are discussed in particular conditions for positivity of as well as for 2 positivity complete positivity optimality and indecomposability are provided in particular we show that for a pair composed of 2 positive and 2 copositive maps there is an indecomposable merging of them one of our main results asserts that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps their canonical merging is an exposed positive map this result provides a wide class of new examples of exposed positive maps as an application new examples of entangled ppt states are described

geometry of quantum systems density states and entanglement various problems concerning the geometry of the space of hermitian operators on a hilbert space are addressed in particular we study the canonical poisson and riemann jordan tensors and the corresponding foliations into k hler submanifolds it is also shown that the space of density states on an n dimensional hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state thus the space of rank k states k 1 n is a smooth manifold of real dimension 2nk k2 1 and this stratification is maximal in the sense that every smooth curve in viewed as a subset of the dual to the lie algebra of the unitary group at every point must be tangent to the strata it crosses for a quantum composite system i e for a hilbert space decomposition an abstract criterion of entanglement is proved

entanglement measure for general pure multipartite quantum states we propose an explicit formula for a measure of entanglement of pure multipartite quantum states we discuss the mathematical structure of the measure and give a brief explanation of its physical motivation we apply the measure on some pure tripartite qubit states and demonstrate that in general the entanglement can depend on what actions are performed on the various subsystems and specifically if the parties in possession of the subsystems cooperate or not we also give some simple but illustrative examples of the entanglement of four qubit and m qubit states

the inverse eigenvalue problem for entanglement witnesses abstract we consider the inverse eigenvalue problem for entanglement witnesses which asks for a characterization of their possible spectra or equivalently of the possible spectra resulting from positive linear maps of matrices we completely solve this problem in the two qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions we also establish a natural duality relationship with the set of absolutely separable states and we completely characterize witnesses i e separating hyperplanes of that set when one of the local dimensions is 2

on fφ hypercentral subgroups of finite groups let g be a finite group we write r g to denote the largest soluble normal subgroup of g and put g r g we say that a chief factor h k of g is non frattini non solubly frattini if h k g k if h k g k respectively a chief factor h k of g is called f central in g provided h k g cg h k f a normal subgroup n of g is said to be f hypercentral f hypercentral in g if either n 1 or n 1 and there exists a chief series 1 n0 n1 nt n of g below n such that every non frattini non solubly frattini respectively factor ni ni 1 of series is f central in g in this paper we analyze some properties and applications of f hypercentral and f hypercentral subgroups

unextendible product bases uncompletable product bases and bound entanglement we report new results and generalizations of our work on unextendible product bases upb uncompletable product bases and bound entanglement we present a new construction for bound entangled states based on product bases which are only completable in a locally extended hilbert space we introduce a very useful representation of a product basis an orthogonality graph using this representation we give a complete characterization of unextendible product bases for two qutrits we present several generalizations of upbs to arbitrary high dimensions and multipartite systems we present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators we prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement

generation of mapping cones from small sets we answer in the affirmative a recently posed question that asked if there exists an untypical convex mapping cone i e one that does not arise from the transpose map and the cones of k positive and k superpositive maps we explicitly construct such a cone based on atomic positive maps our general technique is to consider the smallest convex mapping cone generated by a single map and we derive several results on such mapping cones we use this technique to also present several other examples of untypical mapping cones including a family of cones generated by spin factors we also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality

separability criterion for three qubit states with a four dimensional norm we give a separability criterion for three qubit states in terms of diagonal and anti diagonal entries this gives us a complete characterization of separability when all the entries are zero except for diagonal and anti diagonals the criterion is expressed in terms of a norm arising from anti diagonal entries we compute this norm in several cases so that we get criteria with which we can decide the separability by routine computations

a class of atomic positive linear maps in matrix algebras we find a large class of atomic positive linear maps between n dimensional matrix algebras and study their properties in connection with complete positivity 2 positivity and schwarz inequality

concurrence classes for general pure multipartite states we propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator valued measure on quantum phase in particular we construct wm class ghzm and ghzm 1 class concurrences for general pure m partite states we give explicit expressions for w3 and ghz3 class concurrences for general pure three partite states and for w4 ghz4 and ghz3 class concurrences for general pure four partite states

facial structures for unital positive linear maps in the two dimensional matrix algebra abstract we completely determine the lattice structure for the faces of the convex set of all unital positive linear maps between the 2 2 matrix algebras

quantum entanglement and geometry of determinantal varieties quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of einstein podolsky and rosen recently it has been realized that quantum entanglement is a key ingredient in quantum computation quantum communication and quantum cryptography in this paper we introduce algebraic sets which are determinantal varieties in the complex projective spaces or the products of complex projective spaces for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations these invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states our construction has applications in the following important topics in quantum information theory 1 separability criterion it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable 2 simulation of hamiltonians it is proved that the simulation of semipositive hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets 3 construction of bound entangled mixed states examples of the entangled mixed states which are invariant under partial transpositions thus ppt bound entanglement are constructed systematically from our new separability criterion

all 2 positive linear maps from m3 to m3 are decomposable abstract following an idea of choi we obtain a decomposition theorem for k positive linear maps from m m c to m n c where 2 k min m n as a consequence we give an affirmative answer to kye s conjecture also solved independently by choi that every 2 positive linear map from m 3 c to m 3 c is decomposable

positive groups on hn are completely positive we prove that an operator generates a positive group on the real space of real or complex hermitian matrices if and only if it is a lyapunov operator in particular it follows that every group of positive operators in fact is a group of completely positive operators

new examples of extremal positive linear maps abstract new families of nonnegative biquadratic forms that have 8 9 or 10 real zeros in p 2 p 2 are constructed these are the first examples with 8 9 or 10 real zeros it is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros our examples show that the upper bound is obtained such biquadratic forms define positive linear maps on real symmetric 3 3 matrices that are not completely positive our constructions are explicit and moreover we are able to determine which of the examples are extremal we extend the examples to positive maps on complex matrices and find families of extreme rays in the cone of positive maps

a class of exposed indecomposable positive maps exposed positive maps in matrix algebras define a dense subset of extremal maps we provide a class of indecomposable positive maps in the algebra of 2n 2n complex matrices with n 2 it is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states

exposed positive maps a sufficient condition exposed positive maps in matrix algebras define a dense subset of extremal maps we provide a sufficient condition for a positive map to be exposed this is an analogue of a spanning property which guarantees that a positive map is optimal we analyze a class of decomposable maps for which this condition is also necessary

three qubit entanglement witnesses with the full spanning properties we construct three qubit entanglement witnesses with the full spanning property by which we mean that all the partial transposes have the spanning properties these witnesses determine faces for separable states whose interior lies in the interior of positive partial transposes ppt states and so they detect sets of ppt entangled states with nonempty interiors these witnesses also detect fully bi separable states as well as all kinds of bi separable pure product states our construction depends on the observation that multi partite entanglement witnesses correspond to positive multi linear maps

product vectors in the ranges of multi partite states with positive partial transposes and permanents of matrices in this paper we consider a system of homogeneous algebraic equations in complex variables and their conjugates which arise naturally from the range criterion for separability of ppt states we examine systematically these equations to get sufficient conditions for the existence of nontrivial solutions this gives us possible upper bounds of ranks of ppt entangled edge states and their partial transposes we will focus on the multi partite cases which are much more delicate than the bi partite cases we use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion

performance limits of compressive sensing based signal classification most of the recent compressive sensing cs literature has focused on sparse signal recovery based on compressive measurements however exact signal recovery may not be required in certain signal processing applications such as in inference problems in this paper we provide performance limits of classification of sparse as well as not necessarily sparse signals based on compressive measurements when signals are not necessarily sparse we show that kullback leibler and chernoff distances between two probability density functions under any two hypotheses are preserved up to a factor of m n with m n length compressive measurements compared to that with n length original measurements when the pdfs of the original length observation vectors exhibit certain properties these results are used to quantify the performance limits in terms of upper and lower bounds on the probability of error in signal classification with m length compressive measurements when the signals of interest are sparse in the standard canonical basis performance limits are derived in terms of lower bounds on the probability of error in classifying sparse signals with any classification rule

discrimination on the grassmann manifold fundamental limits of subspace classifiers we derive fundamental limits on the reliable classification of linear and affine subspaces from noisy linear features drawing an analogy between discrimination among subspaces and communication over vector wireless channels we define two shannon inspired characterizations of asymptotic classifier performance first we define the classification capacity which characterizes the necessary and sufficient conditions for vanishing misclassification probability as the signal dimension the number of features and the number of subspaces to be discriminated all approach infinity second we define the diversity discrimination tradeoff which by analogy with the diversity multiplexing tradeoff of fading vector channels characterizes relationships between the number of discernible subspaces and the misclassification probability as the feature noise power approaches zero we derive upper and lower bounds on these quantities which are tight in many regimes numerical results including a face recognition application validate the results in practice

everything you always wanted to know about locc but were afraid to ask in this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication locc while locc emerges as the natural class of operations in many important quantum information tasks its mathematical structure is complex and difficult to characterize here we provide a precise description of locc and related operational classes in terms of quantum instruments our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds while the set of locc is not topologically closed we show that finite round locc constitutes a compact subset of quantum operations additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of locc implementable maps finally we demonstrate a two qubit map whose action can be approached arbitrarily close using locc but nevertheless cannot be implemented perfectly

on the structure of positive maps ii low dimensional matrix algebras we use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the differences and similarities between unital positive maps on m2 c and m3 c our main tool stems from classical grothendieck theorem on tensor product of banach spaces and is an older and more general version of choi jamiolkowski isomorphism between positive maps and block positive choi matrices it takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps in this setting we show that in m2 c case a large class of nice positive maps can be generated from the small set of maps represented by self adjoint unitaries 2px with x maximally entangled vector and p 1 with p rank 1 projector we indicate problems with passing this result to m3 c case among similarities in both m2 c and m3 c cases any unital positive map represented by self adjoint unitary is unitarily equivalent to the transposition map consequently

separability problem for multipartite states of rank at most 4 one of the most important problems in quantum information is the separability problem which asks whether a given quantum state is separable we investigate multipartite states of rank at most 4 which are ppt i e all their partial transposes are positive semidefinite we show that any ppt state of rank 2 or 3 is separable and has length at most 4 for separable states of rank 4 we show that they have length at most 6 it is six only for some qubit qutrit or multiqubit states it turns out that any ppt entangled state of rank 4 is necessarily supported on a 3 3 or a 2 2 2 subsystem we obtain a very simple criterion for the separability problem of the ppt states of rank at most 4 such a state is entangled if and only if its range contains no product vectors this criterion can be easily applied since a four dimensional subspace in the 3 3 or 2 2 2 system contains a product vector if and only if its plucker coordinates satisfy a homogeneous polynomial equation the chow form of the corresponding segre variety we have computed an explicit determinantal expression for the chow form in the former case while such an expression was already known in the latter case

a family of indecomposable positive linear maps based on entangled quantum states abstract we introduce a new family of indecomposable positive linear maps based on entangled quantum states central to our construction is the notion of an unextendible product basis the construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension

the structural physical approximations and optimal entanglement witnesses we introduce the notions of positive and copositive types for entanglement witnesses depending on the distance to the positive part and copositive part an entanglement witness w is of positive type if and only if its partial transpose w is of copositive type we show that if the structural physical approximation spa of w is separable then w should be of copositive type and the spa of w is never separable unless w is of both positive and copositive types this shows that the spa conjecture is meaningful only for those of copositive type we provide examples to show that the spa conjecture fails even for the case of copositive types

unextendible product bases and bound entanglement an unextendible product basis upb for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state we give examples of upbs and show that the uniform mixed state over the subspace complementary to any upb is a bound entangled state we exhibit a tripartite 2 3 2 3 2 upb whose complementary mixed state has tripartite entanglement but no bipartite entanglement i e all three corresponding 2 3 4 bipartite mixed states are unentangled we show that members of a upb are not perfectly distinguishable by local positive operator valued measurements and classical communication s0031 9007 99 09360 6

on the convex cones arising from classifications of partial entanglement in the three qubit system in order to classify partial entanglement of multi partite states it is natural to consider the convex hulls intersections and differences of basic convex cones obtained from partially separable states with respect to partitions of systems in this paper we consider convex cones consisting of x shaped three qubit states arising in this way the class of x shaped states includes important classes like greenberger horne zeilinger diagonal states we find all the extreme rays of those convex cones to exhibit corresponding partially separable states we also give characterizations for those cones which give rise to necessary criteria in terms of diagonal and anti diagonal entries for general three qubit states

a note on the optimality of decomposable entanglement witnesses and completely entangled subspaces entanglement witnesses ews constitute one of the most important entanglement detectors in quantum systems nevertheless their complete characterization in particular with respect to the notion of optimality is still missing even in the decomposable case here we show that for any qubit qunit decomposable ew dew w the three statements are equivalent i the set of product vectors obeying e f w e f 0 spans the corresponding hilbert space ii w is optimal and iii w q with q denoting a positive operator supported on a completely entangled subspace ces and standing for the partial transposition while implications i ii and ii iii are known here we prove that iii implies i this is a consequence of a more general fact saying that product vectors orthogonal to any ces in span after partial conjugation the whole space on the other hand already in the case of the hilbert space there exist dews for which iii does not imply i consequently either i does not imply ii or ii does not imply iii and the above transparent characterization obeyed by qubit qunit dews does not hold in general

classification of bi qutrit positive partial transpose entangled edge states by their ranks we construct 3 3 ppt entangled edge states with maximal ranks to complete the classification of 3 3 ppt entangled edge states by their types the ranks of the states and their partial transposes are 8 and 6 respectively these examples also disprove claims in the literature

concurrence classes for an arbitrary multi qubit state based on a positive operator valued measure in this paper we propose concurrence classes for an arbitrary multi qubit state based on orthogonal complement of a positive operator valued measure or povm for short on quantum phase in particular we construct concurrence for an arbitrary two qubit state and concurrence classes for the three and four qubit states finally we construct wm and ghzm class concurrences for multi qubit states the unique structure of our povm enables us to distinguish different concurrence classes for multi qubit states

positive maps positive polynomials and entanglement witnesses we link the study of positive quantum maps block positive operators and entanglement witnesses with problems related to multivariate polynomials for instance we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares although the general problem of describing the set of positive maps remains open in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity

a class of positive atomic maps we construct a new class of positive indecomposable maps in the algebra of d d complex matrices these maps are characterized by the weakest positivity property and for this reason they are called atomic this class provides a new rich family of atomic entanglement witnesses which define an important tool for investigating quantum entanglement it turns out that they are able to detect states with the weakest quantum entanglement

a new subspace identification algorithm for high resolution doa estimation in this paper we propose a new direction of arrival doa estimator for sensor array processing the estimator is based on a linear algebraic connection between the standard subspace model of the array correlation matrix and a special signal plus interference model which we develop in this paper the estimator we propose is a signal subspace scaled music algorithm which we call ssmusic it is not a subspace weighted music because the scaling depends on the eigenstructure of the estimated signal subspace ssmusic has the advantage of simultaneously estimating the doa and the power of each source we employ a second order perturbation analysis of the estimator and derive stochastic representations for its bias and squared error we compare the new doa estimator with the music estimator based on these representations numerical results demonstrate the superior performance of ssmusic relative to music and the validity of the perturbation results