multiqubit nonlocality in families of 3 and 4 qubit entangled states we investigate genuine multiqubit nonlocality in families of entangled 3 and 4 qubit pure states by analyzing a bell type inequality that is violated only if all qubits are nonlocally correlated we present detailed numerical studies of the relationship between entanglement and violation of the svetlichny bell type inequality in an experimentally accessible set of 3 qubit pure states and identify the special nonlocality property of the maximal slice states in the space of all 3 qubit pure states we also analyze nonlocal correlations in 3 qubit generalized greenberger horne zeilinger ghz states and extend our analysis to the case of 4 qubit generalized ghz states we show that like the 3 qubit case some 4 qubit generalized ghz states do not violate a bell inequality that tests for genuine 4 qubit nonlocality furthermore the location of the boundary between the states that do violate the inequality and those that do not is the same for the 3 and 4 qubit generalized ghz states

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

linear preservers of weak majorization on l1 i when i is an infinite set abstract the necessary and sufficient conditions that a bounded linear map on the banach space l 1 i may be considered as a linear preserver of weak majorization on l 1 i where i is an arbitrary infinite set are given also we prove that the set of all linear preservers of weak majorization is closed under the norm topology

development of omni directional correlation functions for nonlinear model validation in the present study a set of first order correlation functions are proposed to examine the quality of a wide class of identified nonlinear models the first order correlation functions defined as omni directional correlation functions are integrated into two concise tests to provide more effective auto and cross model error correlation diagnosis than the other approaches proposed from higher order correlation functions the mechanisms of the novel validity tests are proved in theory and demonstrated with numerical analyses two simulated case studies in the situation of incorrectly detected model structure and estimated parameters are presented to illustrate the diagnostic power of the new methodology

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

majorization and a schur horn theorem for positive compact operators the nonzero kernel case abstract schur horn theorems focus on determining the diagonal sequences obtainable for an operator under all possible basis changes formally described as the range of the canonical conditional expectation of its unitary orbit following a brief background survey we prove an infinite dimensional schur horn theorem for positive compact operators with infinite dimensional kernel one of the two open cases posed recently by kaftal weiss there they characterized the diagonals of operators in the unitary orbits for finite rank or zero kernel positive compact operators here we show how the characterization problem depends on the dimension of the kernel when it is finite or infinite dimensional we obtain exact majorization characterizations of the range of the canonical conditional expectation of the unitary orbits of positive compact operators with infinite dimensional kernel unlike the approximate characterizations of arveson kadison but extending the exact characterizations of gohberg markus and kaftal weiss recent advances in this subject and related subjects like traces on ideals show the relevance of new kinds of sequence majorization as in the work of kaftal weiss e g strong majorization and another majorization similar to what here we call p majorization and of kalton sukochev e g uniform hardy littlewood majorization and of bownik jasper e g riemann and lebesgue majorization likewise key tools here are new kinds of majorization which we call p and approximate p majorization 0 p

traces ideals and arithmetic means this article grew out of recent work of dykema figiel weiss and wodzicki commutator structure of operator ideals which inter alia characterizes commutator ideals in terms of arithmetic means in this paper we study ideals that are arithmetically mean am stable am closed am open soft edged and soft complemented we show that many of the ideals in the literature possess such properties we apply these notions to prove that for all the ideals considered the linear codimension of their commutator space the number of traces on the ideal is either 0 1 or we identify the largest ideal which supports a unique nonsingular trace as the intersection of certain lorentz ideals an application to elementary operators is given we study properties of arithmetic mean operations on ideals e g we prove that the am closure of a sum of ideals is the sum of their am closures we obtain cancellation properties for arithmetic means for principal ideals a necessary and sufficient condition for first order cancellations is the regularity of the generator for second order cancellations sufficient conditions are that the generator satisfies the exponential 2 condition or is regular we construct an example where second order cancellation fails thus settling an open question we also consider cancellation properties for inclusions and we find and use lattice properties of ideals associated with the existence of gaps

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

multipartite generalization of the schmidt decomposition we find a canonical form for pure states of a general multipartite system in which the constraints on the coordinates with respect to a factorizable orthonormal basis are simply that certain ones vanish and certain others are real for identical particles they are invariant under permutations of the particles as an application we find the dimension of the generic local equivalence class

trace class operators and commutators abstract we identify the linear span of commutators ab ba where a is a trace class operator and b is any bounded operator on a separable hilbert space or where a b are both hilbert schmidt operators thus answering questions raised by work of anderson vaserstein and weiss

the geometry of concurrence as a measure of entanglement it is widely recognized that concurrence can be regarded as a measure of two qubit entanglement this paper studies the geometry of concurrence for a two qubit system to show that the concurrence serves as a coordinate of the factor space where is the space of normalized two qubit states and where g u 1 su 2 su 2 any monotonically increasing function of the concurrence can serve as a measure for entanglement from the viewpoint of riemannian geometry a state with concurrence r is shown to be distant from the separable states by where r ranges over 0 r 1 in addition measures of entanglement for n qubits are discussed on the basis of a bipartite decomposition with m n they are invariant under the local unitary transformation group u 2 u 2m

diagnostic testing of univariate time series models summary the portmanteau statistic for testing the adequacy of an autoregressive moving average model is based on the first m autocorrelations of the residuals from the fitted model this paper examines the properties of this test for various choices of m a modification which allows the use of small values of m is shown to result in a more powerful test the lagrange multiplier statistic godfrey 1979 and a test statistic examined by newbold 1980 are also discussed

errors in variables identification in dynamic networks consistency results for an instrumental variable approach in this paper we consider the identification of a linear module that is embedded in a dynamic network using noisy measurements of the internal variables of the network this is an extension of the errors in variables eiv identification framework to the case of dynamic networks the consequence of measuring the variables with sensor noise is that some prediction error identification methods no longer result in consistent estimates the method developed in this paper is based on a combination of the instrumental variable philosophy and closed loop prediction error identification methods and leads to consistent estimates of modules in a dynamic network we consider a flexible choice of which internal variables need to be measured in order to identify the module of interest this allows for a flexible sensor placement scheme we also present a method that can be used to validate the identified model

geometry of the three qubit state entanglement and division algebras we present a generalization to three qubits of the standard bloch sphere representation for a single qubit and of the seven dimensional sphere representation for two qubits presented in mosseri et al mosseri r and dandoloff r 2001 j phys a math gen 34 10243 the hilbert space of the three qubit system is the 15 dimensional sphere s15 which allows for a natural last hopf fibration with s8 as base and s7 as fibre a striking feature is as in the case of one and two qubits that the map is entanglement sensitive and the two distinct ways of un entangling three qubits are naturally related to the hopf map we define a quantity that measures the degree of entanglement of the three qubit state conjectures on the possibility of generalizing the construction for higher qubit states are also discussed

tripartite entanglement versus tripartite nonlocality in three qubit greenberger horne zeilinger class states we analyze the relationship between tripartite entanglement and genuine tripartite nonlocality for three qubit pure states in the greenberger horne zeilinger class we consider a family of states known as the generalized greenberger horne zeilinger states and derive an analytical expression relating the three tangle which quantifies tripartite entanglement to the svetlichny inequality which is a bell type inequality that is violated only when all three qubits are nonlocally correlated we show that states with three tangle less than 1 2 do not violate the svetlichny inequality on the other hand a set of states known as the maximal slice states does violate the svetlichny inequality and exactly analogous to the two qubit case the amount of violation is directly related to the degree of tripartite entanglement we discuss further interesting properties of the generalized greenberger horne zeilinger and maximal slice states

on local invariants of pure three qubit states we study invariants of three qubit states under local unitary transformations i e functions on the space of entanglement types which is known to have dimension six we show that there is no set of six algebraically independent polynomial invariants of degree 6 and find such a set with maximum degree eight we describe an intrinsic definition of a canonical state on each orbit and discuss the non polynomial invariants associated with it

an infinite dimensional schur horn theorem and majorization theory the main result of this paper is the extension of the schur horn theorem to infinite sequences for two nonincreasing nonsummable sequences and that converge to 0 there exists a positive compact operator a with eigenvalue list and diagonal sequence if and only if j 1n j j 1n j for every n if and only if q for some orthostochastic matrix q when and are summable requiring additionally equality of their infinite series obtains the same conclusion extending a theorem by arveson and kadison our proof depends on the construction and analysis of an infinite product of t transform matrices

an infinite dimensional version of the schur horn convexity theorem the schur horn convexity theorem states that forain rnp u diag a u u u n conv sna wherepdenotes the projection on the diagonal in this paper we generalize this result to the setting of arbitrary separable hilbert spaces it turns out that the theorem still holds if we take thel closure on both sides we will also give a description of the left hand side for nondiagonalizable hermitian operators in the last section we use this result to get an extension theorem for invariant closed convex subsets of the diagonal operators

quantum entanglement the unitary 8 vertex braid matrix with imaginary rapidity we study quantum entanglements induced on product states by the action of 8 vertex braid matrices rendered unitary with purely imaginary spectral parameters rapidity the unitarity is displayed via the canonical factorization of the coefficients of the projectors spanning the basis this adds one more new facet to the famous and fascinating features of the 8 vertex model the double periodicity and the analytic properties of the elliptic functions involved lead to a rich structure of the 3 tangle quantifying the entanglement we thus explore the complex relationship between topological and quantum entanglement

local symmetry properties of pure three qubit states entanglement types of pure states of three spin particles are classified by means of their stabilizers in the group of local unitary transformations it is shown that the stabilizer is generically discrete and that a larger stabilizer indicates a stationary value for some local invariant we describe all the exceptional states with enlarged stabilizers

nonlinear model validation using correlation tests new higher order correlation tests which use model residuals combined with system inputs and outputs are presented to check the validity of a general class of nonlinear models the new method is illustrated by testing both simple and complex nonlinear system models

on covariance function tests used in system identification abstract tests on the autocovariance function of the residuals of an estimated model or on the cross covariance function between the model residuals and the past values of the input signal are frequently used in system identification for model validation purposes usually the threshold of such a test is derived under the assumption that the model and the identified system coincide the present paper shows that this simplifying assumption leads to a risk of overfitting less than expected and accordingly to a greater risk of underfitting the correct test threshold is problem dependent and cannot be determined easily in the general case however it can be shown to lie in a certain interval and to be equal to one of the bounds defining this interval under certain sufficient conditions the implication of these results for model validation based on covariance tests is discussed and illustrated by means of some numerical examples

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

the pythagorean theorem ii the infinite discrete case the study of the pythagorean theorem and variants of it as the basic result of noncommutative metric euclidean geometry is continued the emphasis in the present article is the case of infinite discrete dimensionality

the geometry of multi qubit entanglement this paper a continuation of a previous one iwai 2007 j phys a math theor 40 1361 studies the geometry of multi qubit entanglement with respect to bipartite partitions an n qubit system c2 n is isomorphic with the linear space of 2l 2m complex matrices where l m n according to the isomorphism the local transformation group u 2l u 2m acts on the space of normalized states in let m and g denote the space of normalized states and the local transformation group acting on m respectively according to the orbit types m is stratified into strata among which a principal stratum and the sets of separable states and maximally entangled states will be identified for a function f c det i cc proves to serve as a measure of entanglement where i denotes the 2l 2l identity matrix the f c attains the minimal and the maximal values respectively on the sets of separable states and maximally entangled states and further takes no extremal values on the principal stratum the f c projects to a function on the factor space g m a naturally defined metric on m also projects to that on g m which serves to measure the distance between the separable states f 1 0 and the states f 1 k of prescribed value k of measure to be precise one has to restrict m to the principal stratum when projecting the metric three and four qubit systems will be studied intensively and then multi qubit systems discussed

von neumann entropy and majorization abstract we consider the properties of the shannon entropy for two probability distributions which stand in the relationship of majorization then we give a generalization of a theorem due to uhlmann extending it to infinite dimensional hilbert spaces finally we show that for any quantum channel one has s s for all quantum states if and only if there exists an isometric operator v such that v v

tripartite entanglement dependence of tripartite non locality in non inertial frames the three tangle dependence of smax max s where s is a svetlichny operator is explicitly derived when one party moves with a uniform acceleration with respect to other parties in a generalized greenberger horne zeilinger and maximally slice states the tangle dependence of smax is also derived implicitly from this dependence we conjecture that multipartite entanglement is not the only physical resource for quantum mechanical multipartite non locality

propagation of a quantum state in a continuum of coupled oscillators and applications to entanglement the exponential hilbert space approach is used to link quantum mechanics in a small system with d dimensional hilbert space with the collective quantum behaviour in a large system comprised of d coupled oscillators with hilbert space h in the large system the expectation value of the mode position operator describes the location of a quantum state within the chain of d oscillators and the expectation value of the mode momentum operator describes the change of the mode position with time a continuum of modes d is also considered and in this case these operators obey the commutation relation a consequence of this is that uncertainties in the location of a state and in its momentum obey an uncertainty relation displacements and squeezing in the mode phase space are discussed for a certain hamiltonian the obey equations of motion which are very similar to those of a harmonic oscillator if the system is in an entangled state the formalism can be used to quantify concepts like the location of entanglement and the speed with which the entanglement propagates

classification of n qubit states with minimum orbit dimension the group of local unitary transformations acts on the space of n qubit pure states decomposing it into orbits in a previous paper we proved that a product of singlet states together with an unentangled qubit for a system with an odd number of qubits achieves the smallest possible orbit dimension equal to 3n 2 for n even and 3n 1 2 for n odd where n is the number of qubits in this paper we show that any state with minimum orbit dimension must be of this form and furthermore such states are classified up to local unitary equivalence by the sets of pairs of qubits entangled in singlets

commutator structure of operator ideals commutator structure of operator ideals ken dykema 1 tadeusz figiel gary weiss and mariusz wodzicki department of mathematics texas a m university college station tx 77843 3368 usa b instytut matematyczny polskiej akademii nauk 81 825 sopot poland department of mathematical sciences university of cincinnati cincinnati oh 45221 0025 usa department of mathematics university of california berkeley ca 94720 3840 usa

on test horizon for model validation by output error we derive necessary and sufficient conditions under which the parameters of the model are identical to those of the true system which is supposed to be linear and of finite dimension these conditions are that the cross correlations between the output error and the input as well as the model output should be asymptotically null in a finite horizon this horizon may be used as a minimum necessary test horizon in model validation

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

local unitary symmetries and entanglement invariants we investigate the relation between local unitary symmetries and entanglement invariants of multi qubit systems the hilbert space of such systems can be stratified in terms of states with different types of symmetry we review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties on a stratum of a non trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement finding these invariants is often a simpler problem than finding the invariants of the local unitary group the conditions as given by the luna richardson theorem for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed as an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non trivial on the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations this canonical form and the invariants are directly determined by the symmetry group further we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties

linear preservers of majorization on lp i abstract in this paper we use doubly stochastic operators to extend the notion of majorization on l p where i is assumed to be an infinite set and p 1 and then characterize the structure of all bounded linear maps on this space which preserve majorization we will see that the two cases p 1 and p 1 lead to different structures