separability of sequential isomorphisms on quantum effects in multipartite systems in the paper we study separability of sequential isomorphisms on quantum effects in multipartite systems first we obtain necessary and sufficient conditions for separable sequential isomorphisms in bipartite systems secondly in finite dimensional bipartite systems separability of sequential endomorphisms on quantum effects is discussed furthermore we extend these conclusions to the multipartite case finally applying our results an equivalent characterization of local unitary operations is given

potential theory for the sum of the 1 laplacian and p laplacian abstract the variational problem of minimizing the functional u d u 1 p d u p a u on a domain r n under zero boundary values which appears for example in the theory of bingham fluids shows interesting phenomena such as the formation of a subset of with positive measure where the minimizer is constant a plateau the corresponding euler lagrange equation should be 1 u p u a but the 1 laplacian is not defined in points where d u 0 we show that it is nevertheless possible to define subsolutions and supersolutions for this equation we show that the comparison principle is valid and we begin developing a potential theory analogous to what is known about the p laplacian in addition we apply those notions to rotationally invariant solutions and we discuss properties of the plateau and a possible connection to the theory

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

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

unique decompositions faces and automorphisms of separable states let sk be the set of separable states on b cm cn admitting a representation as a convex combination of k pure product states or fewer if m 1 n 1 and k max m n we show that sk admits a subset vk such that vk is dense and open in sk and such that each state in vk has a unique decomposition as a convex combination of pure product states and we describe all possible convex decompositions for a set of separable states that properly contains vk in both cases we describe the associated faces of the space of separable states which in the first case are simplexes and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras as an application of these results we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of b cm cn that preserve entanglement and separability

regularity of quasi minimizers for non uniformly elliptic integrals abstract in this paper we consider a class of non uniformly elliptic integral functionals and we prove the local boundedness of the quasi minimizers our approach is based on a suitable adaptation of the celebrated de giorgi proof and it relies on an appropriate caccioppoli type inequality

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

quantum measurements and maps preserving strict convex combinations and pure states in this paper a characterization of maps between quantum states that preserve pure states and strict convex combinations is obtained based on this characterization a structure theorem for maps between multipartite quantum states that preserve separable pure states and strict convex combinations is established then these results are applied to characterize injective local quantum measurements and answer some conjectures proposed in hou and liu 2012 j phys a math theor 45 205305

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

minimizing non convex multiple integrals a density result we consider variational problems of the form where is a bounded open set in r n f r n r is a possibly non convex lower semicontinuous function with p growth at infinity for some 1 p u 0 is any function in w 1 p assuming that the convex envelope f of f is affine on each connected component of the set f f we prove the existence of solutions to p for every continuous function g such that i g has no strict local minima and ii every convergent sequence of extremum points of g eventually belongs to an interval where g is constant thus showing that the set of continuous functions g that yield existence to p is dense in the space of continuous functions on r

local regularity for solutions of anisotropic obstacle problems abstract this paper deals with anisotropic obstacle problems for the a harmonic equation div a x u x 0 a local regularity result is given under suitable assumptions

finding decompositions of a class of separable states abstract we consider the class of separable states which admit a decomposition i a i b i with the b i s having independent images we give a simple intrinsic characterization of this class of states given a density matrix in this class we construct such a decomposition which can be chosen so that the a i s are distinct with unit trace and then the decomposition is unique we relate this to the facial structure of the set of separable states the states investigated include a class that corresponds under the choi jamiolkowski isomorphism to the quantum channels called quantum classical and classical quantum by holevo

on the nash point equilibria in the calculus of variations abstract the existence and partial regularity of the nash point equilibria for a pair of multiple integrals j u v f x u v u v dx k u v g x u v u v dx are studied the conditions as well as the results are similar to those for local minima obtained by acerbi fusco and giaquinta giusti

stability for the solutions of div u p 2 u f with varying p abstract elliptic partial differential equations with principal part div u p 2 u are applied in physics e g for the description of phenomena in glaceology the objective of our note is a natural question of stability for solutions as p varies

partial regularity of p x harmonic maps let g x and hij u be uniformly elliptic symmetric matrices and assume that hij u and p x 2 are sufficiently smooth we prove partial regularity of minimizers for the functional f u z

existence and boundedness of a minimiser for a constrained minimisation problem on rn with limiting exponent we use the concentration compactness principle to study the existence of a minimiser of the minimisation problem where u u 1 u n we also prove the boundedness of the minimiser of l 1 by using the reverse holder inequality

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

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

on the blow up of heat flow for conformal 3 harmonic maps using a comparison theorem chang ding and ye 1992 proved a finite time derivative blow up for the heat flow of harmonic maps from d 2 a unit ball in r 2 to s 2 a unit sphere in r 3 under certain initial and boundary conditions we generalize this result to the case of 3 harmonic map heat flow from d 3 to s 3 in contrast to the previous case our governing parabolic equation is quasilinear and degenerate technical issues such as the development of a new comparison theorem have to be resolved

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

entropy preserving maps on quantum states abstract in the paper we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations let s h be the set of all quantum states positive operators with the unit trace on a complex hilbert space h with dim h n and s the quantum entropy of the quantum state for arbitrary a surjective map s h s h we claim that preserves quantum entropy of convex combinations of quantum states i e satisfies s t 1 t s t 1 t for arbitrary s h and t 0 1 if and only if there exists a unitary or anti unitary operator u on h such that u u for all s h

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

liouville theorems partial regularity and holder continuity of weak solutions to quasilinear elliptic systems this paper describes the connections between liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth it is shown that equivalence does in general hold only in some restricted sense a complete correspondence can be established in certain cases e g for small solutions and for minima of quadratic variational integrals

automorphisms of even unimodular lattices and unramified salem numbers abstract in this paper we study the characteristic polynomials s x det xi f iip q of automorphisms of even unimodular lattices with signature p q in particular we show that any salem polynomial of degree 2n satisfying s 1 s 1 1 n arises from an automorphism of an indefinite lattice a result with applications to k3 surfaces

removability of singular sets of harmonic maps it is proved that a harmonic map with small energy and the monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density moreover the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability this work generalizes the results in cl df lg12 which were proved under the assumption that the singular sets are isolated points or smooth submanifolds

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

global analysis of a p ginzburg landau energy with radial structure abstract this paper is concerned with the asymptotic behavior of a p ginzburg landau functional with radial structure as parameter goes to zero in the case of p 2 by analyzing the functional globally we show that the singularity of p ginzburg landau energy concentrates on the origin by the fact the singularity can be balanced by some infinitesimal weight we prove that an energy with a proper weight is globally bounded

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

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

conditions for uniqueness of product representations for separable quantum channels and separable quantum states we give a sufficient condition that an operator sum representation of a separable quantum channel in terms of product operators is the unique product representation for that channel and then provide examples of such channels for any number of parties this result has implications for efforts to determine whether or not a given separable channel can be exactly implemented by local operations and classical communication by the choi jamiolkowski isomorphism it also translates to a condition for the uniqueness of product state ensembles representing a given quantum state these ideas follow from considerations concerning whether or not a subspace spanned by a given set of product operators contains at least one additional product operator

subelliptic harmonic maps we study a nonlinear harmonic map type system of subelliptic pde in particular we solve the dirichlet problem with image contained in a convex ball

interior w2 p x regularity for elliptic equations of nondivergence form abstract in this paper we consider the following elliptic equations of nondivergence form i j 1 n a i j 2 u x i x j f in r n where the coefficients a i j are real valued bounded measurable functions defined in r n belong to vmo loc see definition 2 5 and satisfy the uniformly elliptic condition under the assumption that p satisfies the log holder continuity condition we will establish the interior w 2 p x regularity for strong solutions to the elliptic equations of nondivergence form

global minimizers for a p ginzburg landau type energy in r 2 given a p 2 we prove existence of global minimizers for a p ginzburg landau type energy over maps on r2 with degree d 1 at infinity for the analogous problem on the half plane we prove existence of a global minimizer when p is close to 2 the key ingredient of our proof is the degree reduction argument that allows us to construct a map of degree d 1 from an arbitrary map of degree d 1 without increasing the p ginzburg landau energy

the classification of purely non symplectic automorphisms of high order on k3 surfaces abstract an automorphism of order n of a k3 surface is called purely non symplectic if it multiplies the holomorphic symplectic form by a primitive n th root of unity we give the classification of purely non symplectic automorphisms with n 12 where denotes the euler totient function

length filtration of the separable states we investigate the separable states of an arbitrary multi partite quantum system with hilbert space formula see text of dimension d the length l of is defined as the smallest number of pure product states having as their mixture the length filtration of the set of separable states formula see text is the increasing chain formula see text where formula see text we define the maximum length formula see text critical length lcrit and yet another special length lc which was defined by a simple formula in one of our previous papers the critical length indicates the first term in the length filtration whose dimension is equal to formula see text we show that in general d lc lcrit lmax d2 we conjecture that the equality lcrit lc holds for all finite dimensional multi partite quantum systems our main result is that lcrit lc for the bipartite systems having a single qubit as one of the parties this is accomplished by computing the rank of the jacobian matrix of a suitable map having formula see text as its range

optimal ensemble length of mixed separable states the optimal pure state ensemble length of a separable state a is the minimum number of pure states needed in convex combination to construct a we study the set of all separable states with optimal pure state ensemble length equal to k or fewer lower bounds on k are found below which these sets have measure 0 in the set of separable states in the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest the lower bounds are sharp a consequence of our results is that for all two particle systems except possibly those with one qubit or those with a nine dimensional hilbert space and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state s rank in bipartite systems with probabilty 1 it is greater than 1 4 the rank raised to the 3 2 power and in a system of p qubits with probability 1 it is greater than 22p 1 2p i e alm

determinants of integral ideal lattices and automorphisms of given characteristic polynomial abstract the aim of this paper is to give a characterisation of the determinants and signatures of integral ideal lattices over a given algebraic number field this is then used to obtain an existence criterion for automorphisms of given characteristic polynomial in particular we give a different proof of a result of b gross and c mcmullen in preparation

ginzburg landau minimizers with prescribed degrees capacity of the domain and emergence of vortices let omega subset mathbb r 2 be a simply connected domain let omega be a simply connected subdomain of omega and set a omega setminus overline omega let j be the class of complex valued maps on the annular domain a with degree 1 both on partial omega and on partial omega we consider the variational problem of minimizing the ginzburg landau energy e lambda among all maps in j because only the degrees of the map are prescribed on the boundary the set j is not closed under weak h 1 convergence we show that the attainability of the minimum of e lambda over j is determined by the value of cap a where cap a is the h 1 capacity of the domain a in contrast it is known that the existence of minimizers of e lambda among the maps with a prescribed dirichlet boundary data does not depend on this geometric characteristic when cap a ge pi such an a is denoted as either subcritical if cap a pi or critical if cap a pi we show that the global minimizers of e lambda exist for each lambda 0 and that they are vortexless when lambda is large assuming that lambda to infty we demonstrate that the minimizers of e lambda converge in h 1 a and even better to an mathbb s 1 valued harmonic map which we explicitly identify when cap a lambda 0 or ii the global minimizers exist for each lambda 0 we conjecture that the second case never occurs further for large lambda we establish that the minimizing sequences minimizers in supercritical domains develop exactly two vortices a vortex of degree 1 near partial omega and a vortex of degree 1 near partial omega

a geometric characterization of invertible quantum measurement maps a geometric characterization is given for invertible quantum measurement maps denote by s h the convex set of all states i e trace 1 positive operators on hilbert space h with dimh and 1 2 the line segment joining two elements 1 2 in s h it is shown that a bijective map s h s h satisfies 1 2 1 2 for any 1 2 s if and only if has one of the following forms m m tr m m or m tm tr m tm where m is an invertible bounded linear operator and t is the transpose of with respect to an arbitrarily fixed orthonormal basis

some regularity results for p harmonic mappings between riemannian manifolds abstract let m be a c 2 smooth riemannian manifold with boundary and n a complete c 2 smooth riemannian manifold we show that each stationary p harmonic mapping u m n whose image lies in a compact subset of n is locally c 1 for some 0 1 provided that n is simply connected and has non positive sectional curvature we also prove similar results for minimizing p harmonic mappings with image being contained in a regular geodesic ball moreover when m has non negative ricci curvature and n is simply connected with non positive sectional curvature we deduce a gradient estimate for c 1 smooth weakly p harmonic mappings from which follows a liouville type theorem in the same setting

linear rank preservers of tensor products of rank one matrices abstract let n 1 n k be integers larger than or equal to 2 we characterize linear maps m n 1 n k m n 1 n k such that r a n k a 1 a k 1 whenever r a n k a 1 a k 1 for all a i m n i i 1 k applying this result we extend two recent results on linear maps that preserve the rank of special classes of matrices

the automorphism group of separable states in quantum information theory we show that the linear group of automorphism of hermitian matrices which preserves the set of separable states is generated by natural automorphisms change of an orthonormal basis in each tensor factor partial transpose in each tensor factor and interchanging two tensor factors of the same dimension we apply our results to the preservers of the product numerical range

generic local distinguishability and completely entangled subspaces a subspace of a multipartite hilbert space is completely entangled if it contains no product states such subspaces can be large with a known maximum size smax approaching the full dimension of the system d we show that almost all subspaces with dimension s smax are completely entangled and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n d smax this condition holds for almost all sets of states of all multipartite systems and reveals something surprising the criterion is identical for separable and nonseparable states entanglement makes no difference

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

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

the regularity of harmonic maps into spheres and applications to bernstein problems we show the regularity of and derive a priori estimates for weakly harmonic maps from a riemannian manifold into a euclidean sphere under the assumption that the image avoids some neighborhood of a half equator the proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems we apply these results to the spherical and euclidean bernstein problems for minimal hypersurfaces obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in euclidean spaces are trivial

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

bipartite states of low rank are almost surely entangled we show that a bipartite state on a tensor product of two matrix algebras is almost surely entangled if its rank is not greater than that of one of its reduced density matrices

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

harnack s inequality for quasiminimizers with nonstandard growth conditions we prove harnack inequalities for quasiminimizers of the variable exponent dirichlet energy integral by employing the de giorgi method

mixed states for an allen cahn type equation this paper continues the recent study of an allen cahn model pde 1 by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions enabling us to obtain multibump solutions in a much more general setting as in 1 novel minimization arguments play a key role in finding solutions

vanishing viscosity limit for homogeneous axisymmetric no swirl solutions of stationary navier stokes equations abstract 1 homogeneous axisymmetric no swirl solutions of three dimensional incompressible stationary navier stokes equations which are smooth on the unit sphere minus the north and south poles have been classified in this paper we study the vanishing viscosity limit of sequences of these solutions as the viscosity tends to zero some sequences of solutions c l o c m converge to solutions of euler equations on the sphere minus the poles while for other sequences of solutions transition layer behaviors occur for every latitude circle there are sequences which c l o c m converge respectively to different solutions of the euler equations on the spherical caps above and below the latitude circle we give detailed analysis of these convergence and transition layer behaviors

algebraic constructions of densest lattices abstract the aim of this paper is to investigate rotated versions of the densest known lattices in dimensions 2 3 4 5 6 7 8 12 and 24 constructed via ideals and free z modules that are not ideals in subfields of cyclotomic fields the focus is on totally real number fields and the associated full diversity lattices which may be suitable for signal transmission over both gaussian and rayleigh fading channels we also discuss on the existence of a number field k such that it is possible to obtain the lattices a 2 e 6 and e 7 via a twisted embedding applied to a fractional ideal of o k