all degree 6 local unitary invariants of k qudits we give explicit index free formulae for all the degree 6 and also degree 4 and 2 algebraically independent local unitary invariant polynomials for finite dimensional k partite pure and mixed quantum states we carry this out using graph technical methods which provide illustrations for this abstract topic

nonbinary quantum codes we present several results on quantum codes over general alphabets that is in which the fundamental units may have more than two states in particular we consider codes derived from finite symplectic geometry assumed to have additional global symmetries from this standpoint the analogs of calderbank shor steane codes and of gf 4 linear codes turn out to be special cases of the same construction this allows us to construct families of quantum codes from certain codes over number fields in particular we get analogs of quadratic residue codes including a single error correcting code encoding one letter in five for any alphabet size we also consider the problem of fault tolerant computation through such codes generalizing ideas of gottesman see phys rev a vol 57 no 1 p127 37 1998

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

a semidefinite program for distillable entanglement we show that the maximum fidelity obtained by a positive partial transpose p p t distillation protocol is given by the solution to a certain semidefinite program this gives a number of new lower and upper bounds on p p t distillable entanglement and thus new upper bounds on 2 locally distillable entanglement in the presence of symmetry the semidefinite program simplifies considerably becoming a linear program in the case of isotropic and werner states using these techniques we determine the p p t distillable entanglement of asymmetric werner states and maximally correlated states we conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1 local distillation

evaluation of a ln tan integral arising in quantum field theory we analytically evaluate a dilogarithmic integral that is prototypical of volumes of ideal tetrahedra in hyperbolic geometry we additionally obtain new representations of the clausen function cl2 and the catalan constant g cl2 2 as well as new relations between sine and clausen function values

badly approximable points on planar curves and winning abstract for any i j 0 with i j 1 let bad i j denote the set of points x y r 2 such that max q x 1 i q y 1 j c q for some positive constant c c x y and all q n we show that bad i j c is winning in the sense of schmidt games for a large class of planar curves c namely everywhere non degenerate planar curves and straight lines satisfying a natural diophantine condition this strengthens recent results solving a problem of davenport from the sixties in short within the context of davenport s problem the winning statement is best possible furthermore we obtain the inhomogeneous generalisations of the winning results for planar curves and lines and also show that the inhomogeneous form of bad i j is winning for two dimensional schmidt games

2 dimensional badly approximable vectors and schmidt s game we prove that for any pair s t of nonnegative numbers with s t 1 the set of s t badly approximable vectors in r 2 is 34 p 2 1 winning for schmidt s game

bounded orbits of diagonalizable flows on finite volume quotients of products of sl2 r abstract we prove a number field analogue of a result of c mcmullen that the set of badly approximable numbers is absolute winning we also prove a weighted version we use this to prove an instance of a conjecture of an guan and kleinbock 4 namely let g sl 2 r sl 2 r and be a lattice in g we show that the set of points on g whose orbits under a one parameter ad semisimple subgroup of g are bounded form a hyperplane absolute winning set

local equivalence of rank 2 quantum mixed states we investigate the equivalence of quantum mixed states under local unitary transformations for a class of rank 2 mixed states a sufficient and necessary condition of local equivalence is obtained by giving a complete set of invariants under local unitary transformations such that two states in this class are locally equivalent if and only if all these invariants have equal values for them

on a three dimensional symmetric ising tetrahedron and contributions to the theory of the dilogarithm and clausen functions perturbative quantum field theory for the ising model at the three loop level yields a tetrahedral feynman diagram c a b with masses a and b and four other lines with unit mass the completely symmetric tetrahedron ctet c 1 1 has been of interest from many points of view with several representations and conjectures having been given in the literature we prove a conjectured exponentially fast convergent sum for c 1 1 as well as a previously empirical relation for c 1 1 as a remarkable difference of clausen function values our presentation includes propositions extending the theory of the dilogarithm li2 and clausen cl2 functions as well as their relation to other special functions of mathematical physics the results strengthen connections between feynman diagram integrals volumes in hyperbolic space number theory and special functions and numbers specifically including dilogarithms clausen function values and harmonic numbers

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

concurrence of arbitrary dimensional bipartite quantum states we derive an analytical lower bound for the concurrence of a bipartite quantum state in arbitrary dimension a functional relation is established relating concurrence the peres horodecki criterion and the realignment criterion we demonstrate that our bound is exact for some mixed quantum states the significance of our method is illustrated by giving a quantitative evaluation of entanglement for many bound entangled states some of which fail to be identified by the usual concurrence estimation method

formulae concerning the computation of the clausen integral cl2 θ the main problem under study concerns the expression of the clausen integral cl2 in closed form in terms of known constants and special functions when is equal to a rational multiple of belonging to 0 2 a general formula giving cl2 p q in terms of the derivative of the di gamma function and the sine function is deduced from an appropriate fourier series expansion some variants of this formula are obtained in further sections the formulae expressing cl2 2 and more generally cl2 m m 2 3 4 as linear combinations of terms of the form cl2 const are established the various results are illustrated by means of typical examples of practical application the last section contains two simple approximations enabling the computation of cl for any in 0 with a relative error smaller than 0 63 and 0 003 resp the paper ends with an appendix in which among other things a peculiar trigonometric identity is established as a by product

polynomial invariants of quantum codes the weight enumerators shor and laflamme 1997 of a quantum code are quite powerful tools for exploring its structure as the weight enumerators are quadratic invariants of the code this suggests the consideration of higher degree polynomial invariants we show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one to one correspondence with s sub k sup n we then present a number of equations and inequalities in these invariants in particular we give a higher order generalization of the shadow enumerator of a code and prove that its coefficients are nonnegative we also prove that the quartic invariants of a 4 4 2 sub 2 code are uniquely determined an important step in a proof that any 4 4 2 sub 2 code is additive rains 1999

tensor network methods for invariant theory invariant theory is concerned with functions that do not change under the action of a given group here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states this graphical approach provides an alternative to the polynomial equations that describe invariants which often contain a large number of terms with coefficients raised to high powers this approach also enables one to use known methods from tensor network theory such as the matrix product state mps factorization when studying polynomial invariants as our main example we consider invariants of mpss we generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the r nyi entropies we find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted as well as an alternative and sometimes much simpler means to study polynomial invariants of quantum states in addition many tensor network methods such as mpss contain excellent tools that can be applied in the study of invariants

badly approximable points on manifolds this paper is motivated by two problems in the theory of diophantine approximation namely davenport s problem regarding badly approximable points on submanifolds of a euclidean space and schmidt s problem regarding the intersections of the sets of weighted badly approximable points the problems have been recently settled in dimension two but remain open in higher dimensions in this paper we develop new techniques that allow us to tackle them in full generality the techniques rest on lattice points counting and a powerful quantitative result of bernik kleinbock and margulis the main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of mathbb r n has full dimension one of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree

a note on equivalence of bipartite states under local unitary transformations the equivalence of arbitrary dimensional bipartite states under local unitary transformations lut is studied a set of invariants and ancillary invariants under lut is presented we show that two states are equivalent under lut if and only if they have the same values for all of these invariants

quantum codes of minimum distance two it is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2 thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize with this in mind we present a number of results on codes of minimum distance 2 we first compute the linear programming bound on the dimension of such a code then show that this bound can only be attained when the code either is of even length or is of length 3 or 5 we next consider questions of uniqueness showing that the optimal code of length 2 or 1 is unique implying that the well known one qubit in five single error correcting code is unique and presenting nonadditive optimal codes of all greater even lengths finally we compute the full automorphism group of the more important distance 2 codes allowing us to determine the full automorphism group of any gf 4 linear code

on a problem in simultaneous diophantine approximation schmidt s conjecture for any i j 0 with i j 1 let bad i j denote the set of points x y 2 r 2 for which maxfkqxk 1 i kqyk 1 j g c q for all q 2 n here c c x y is a positive constant our main result implies that any nite intersection of such sets has full dimension this settles a conjecture of wolfgang m schmidt in the theory of simultaneous diophantine approximation

the dilogarithm function for complex argument this paper summarizes the basic properties of the euler dilogarithm function often referred to as the spence function these include integral representations series expansions linear and quadrat

quantum weight enumerators in a recent paper shor and laflamme see phys rev lett vol 78 p 1600 2 1997 defined two weight enumerators for quantum error correcting codes connected by a macwilliams transform and used them to give a linear programming bound for quantum codes we introduce two new enumerators which while much less powerful at producing bounds are useful tools nonetheless the new enumerators are connected by a much simpler duality transform clarifying the duality between shor and laflamme s enumerators we also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance and to extend the enumerator theory to codes with block size greater than 2

on the clausen integral cl2 θ and a related integral abstract in this article the clausen integral cl 2 0 ln 2 sin t 2 d t whereby is equal to a rational multiple of belonging to 0 2 is developed into a finite series of derivatives of the di gamma function for the same values of a method is presented to calculate the related integral 1 0 ln 2 2 cos 1 d

applications of the duality method to generalizations of the jordan canonical form abstract the jordan normal form for a matrix over an arbitrary field and the canonical form for a pair of matrices under contragredient equivalence are derived using ptak s duality method

flows on homogeneous spaces and diophantine properties of matrices we generalize the notions of badly approximable resp singular systems of m linear forms in n variables and relate these generalizations to certain bounded resp divergent trajectories in the space of lattices in rm n