algebraic bethe ansatz for the gl 1 2 generalized model ii the three gradings the algebraic bethe ansatz can be performed rather abstractly for whole classes of models sharing the same r matrix the only prerequisite being the existence of an appropriate pseudo vacuum state here we perform the algebraic bethe ansatz for all models with 9 9 rational gl 1 2 invariant r matrix and all three possibilities of choosing the grading our bethe ansatz solution applies for instance to the supersymmetric t j model the supersymmetric u model and a number of interesting impurity models it may be extended to obtain the quantum transfer matrix spectrum for this class of models the properties of a specific model enter the bethe ansatz solution i e the expression for the transfer matrix eigenvalue and the bethe ansatz equations through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model

on form factors in nested bethe ansatz systems we investigate form factors of local operators in a multi component quantum nonlinear schrodinger model a prototype theory solvable by the so called nested bethe ansatz we determine the analytic properties of the infinite volume form factors using the coordinate bethe ansatz solution and establish a connection with the finite volume matrix elements in the two component models we derive a set of recursion relations for the magnonic form factors which are the matrix elements on the nested bethe ansatz states in certain simple cases involving states with only one spin impurity we obtain explicit solutions for the recursion relations

on almost randomizing channels with a short kraus decomposition for large d we study quantum channels on cd obtained by selecting randomly n independent kraus operators according to a probability measure on the unitary group mathcal u d when is the haar measure we show that for n succcurlyeq d varepsilon 2 such a channel is e randomizing with high probability which means that it maps every state within distance e d in operator norm of the maximally mixed state this slightly improves on a result by hayden leung shor and winter by optimizing their discretization argument

scalar products of bethe vectors in models with mathfrak gl 2 1 symmetry 1 super analog of reshetikhin formula we study scalar products of bethe vectors in integrable models solvable by nested algebraic bethe ansatz and possessing mathfrak gl 2 1 symmetry using explicit formulas of the monodromy matrix entries multiple actions onto bethe vectors we obtain a representation for the scalar product in the most general case this explicit representation appears to be a sum over partitions of the bethe parameters it can be used for the analysis of scalar products involving on shell bethe vectors as a by product we obtain a determinant representation for the scalar products of generic bethe vectors in integrable models with mathfrak gl 1 1 symmetry

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

supersymmetric vertex models with domain wall boundary conditions by means of the drinfeld twists we derive the determinant representations of the partition functions for the gl 1 1 and gl 2 1 supersymmetric vertex models with domain wall boundary conditions in the homogeneous limit these determinants degenerate to simple functions

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

prime factorization using square root approximation many cryptosystems are based on the factorization of large integers the complexity of this type of factorization is still an advantage for data security developers and a challenge for both mathematicians and cryptanalysts the security of rsa relies on the difficulty of factoring large integers the factorization was studied earlier by old civilizations like the greek but their methods were extended after the emergence of computers the paradox of rsa is that in order to make rsa more efficient we use a modulus n pq which is as small as possible on the other hand it is sufficient to factor n in order to decrypt the encrypted messages in this paper we propose a new factorization method based on the square root approximation this method allows in reducing the search for candidate prime factors of a given integer by approximating each prime factor before considering it as a candidate

determinant representations of correlation functions for the supersymmetric t j model working in the f basis provided by the factorizing f matrix the scalar products of bethe states for the supersymmetric t j model are represented by determinants by means of these results we obtain determinant representations of correlation functions for the model

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

volume ratio sparsity and minimaxity under unitarily invariant norms this paper studies non asymptotic minimax estimation of high dimensional matrices and provides tight minimax rates for a large collection of loss functions in a variety of problems via information theoretic methods based on the convex geometry of finite dimensional banach spaces we first develop a volume ratio approach for determining minimax estimation rates of unconstrained mean matrices under all unitarily invariant norm losses which turn out to only depend on the norm of identity matrix in addition we establish the minimax rates for estimating normal mean matrices with submatrix sparsity where the sparsity constraint introduces an additional term in the rate which in contrast to the unconstrained case is determined by the smoothness lipschitz constant of the norm this method is also applicable to the low rank matrix completion problem and extends well beyond the additive noise model in particular it yields tight rates in covariance matrix estimation and poisson rate matrix estimation problems for all unitarily invariant norms

integral representations for correlation functions of the xxz chain at finite temperature we derive a novel multiple integral representation for a generating function of the z z correlation functions of the spin xxz chain at finite temperature and finite longitudinal magnetic field our work combines algebraic bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics

the quantum inverse scattering method approach to correlation functions the inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with ther matrix of xxx type these models include the one dimensional bose gas and the heisenberg xxx model the algebraic questions of the problem are considered

norms and scalar products for ads 3 we compute scalar products and norms of bethe vectors in the massless sector of ads3 integrable superstring theories by exploiting the general difference form of the s matrix of massless excitations in the pure ramond ramond case and the difference form valid only in the bmn limit in the mixedflux case we obtain determinant like formulas for the scalar products generalising a procedure developed in previous literature for standard r matrices to the present non conventional situation we verify our expressions against explicit calculations using bethe vectors for chains of small length and perform some computer tests of the exact formulas as far as numerical accuracy sustains us this should be the first step towards the derivation of integrable form factors and correlation functions for the ads3 s matrix theory

von neumann entropy penalization and low rank matrix estimation we study a problem of estimation of a hermitian nonnegatively definite matrix of unit trace for instance a density matrix of a quantum system based on n i i d measurements x1 y1 xn yn where

scalar products of bethe vectors in models with gl 2 1 symmetry 2 determinant representation we study integrable models with mathfrak gl 2 1 symmetry and solvable by nested algebraic bethe ansatz we obtain a determinant representation for scalar products of bethe vectors when the bethe parameters obey some relations weaker than the bethe equations this representation allows us to find the norms of on shell bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix

interaction in quantum communication in some scenarios there are ways of conveying information with many fewer even exponentially fewer qubits than possible classically moreover some of these methods have a very simple structure they involve only few message exchanges between the communicating parties it is therefore natural to ask whether every classical protocol may be transformed to a simpler quantum protocol one that has similar efficiency but uses fewer message exchanges we show that for any constant k there is a problem such that its k 1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity this in particular proves a round hierarchy theorem for quantum communication complexity and implies via a simple reduction an omega n1k lower bound for k message quantum protocols for set disjointness for constant k enroute we prove information theoretic lemmas and define a related measure of correlation the informational distance that we believe may be of significance in other contexts as well

optimal estimation of low rank density matrices the density matrices are positively semi definite hermitian matrices of unit trace that describe the state of a quantum system the goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography in particular in the case of pauli measurements with explicit dependence of the bounds on the rank and other complexity parameters such bounds are established for several statistically relevant distances including quantum versions of kullback leibler divergence relative entropy distance and of hellinger distance so called bures distance and schatten p norm distances sharp upper bounds and oracle inequalities for least squares estimator with von neumann entropy penalization are obtained showing that minimax lower bounds are attained up to logarithmic factors for these distances

integral representation of the density matrix of the xxz chain at finite temperatures we present an integral formula for the density matrix of a finite segment of the infinitely long spin xxz chain this formula is valid for any temperature and any longitudinal magnetic field