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Shmuel Friedland

Publications and source records attributed to Shmuel Friedland.

At least 73 records · Page 4Linked to original sources
arXiv2012

Upper bounds on the magnitude of solutions of certain linear systems with integer coefficients

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some positive integer $k$. We show that if the system has a nontrivial solution then there exists a nontrivial solution $\x=(x_1,...,x_n)\trans$ such that $\frac{|x_j|}{|x_i|}\le k^{n-1}$ for each $i,j$ satisfying $x_ix_j\ne 0$. This inequality is sharp. We also prove a conjecture of A. Tyszka related to our results.

math.CA↗
arXiv2012

Results and open problems in matchings in regular graphs

This survey paper deals with upper and lower bounds on the number of $k$-matchings in regular graphs on $N$ vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its variants. We then discuss non-bipartite graphs. Little is known beyond the recent proof of the Lovász-Plummer conjecture on the exponential growth of perfect matchings in cubic bridgeless graphs. We discuss the problem of the minimum of haffnians on the convex set of matrices, whose extreme points are the adjacency matrices of subgraphs of the complete graph corresponding to perfect matchings. We also consider infinite regular graphs. The analog of $k$-matching is the $p$-monomer entropy, where $p\in [0,1]$ is the density of the number of matchings.

math.CO↗
arXiv2011

The minimum entropy output of a quantum channel is locally additive

We show that the minimum von-Neumann entropy output of a quantum channel is locally additive. Hasting's counterexample for the additivity conjecture, makes this result quite surprising. In particular, it indicates that the non-additivity of the minimum entropy output is a global effect of quantum channels.

quant-ph↗
arXiv2011

Local extrema of entropy functions under tensor products

We show that under a certain condition of local commutativity the minimum von-Neumann entropy output of a quantum channel is locally additive. We also show that local minima of the 2-norm entropy functions are closed under tensor products if one of the subspaces has dimension 2.

math-ph↗
arXiv2011

A proof of the set-theoretic version of the salmon conjecture

We show that the irreducible variety of 4 x 4 x 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials), and of degree 9 (the symmetrization conditions).

math.AG↗
arXiv2011

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 \emph{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 preservers of the product numerical range.

quant-ph↗
arXiv2011

A note on the nonzero spectrum of irreducible matrices

In this note we extend the necessary and sufficient conditions of Boyle-Handleman 1991 and Kim-Ormes-Roush 2000 for a nonzero eigenvalue multiset of primitive matrices over $\R_+$ and $\Z_+$, respectively, to irreducible matrices.

math.SP↗
arXiv2011

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).

math-ph↗
arXiv2011

On the generic and typical ranks of 3-tensors

We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n not greater than 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m, n=(m-1)^2+1, m=3,4,..., which have at least two typical ranks.

math.AG↗
arXiv2010

Closed formula for the relative entropy of entanglement in all dimensions

The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find a closed formula for all the entangled state for which this state is a CSS. Quite amazing, our formula holds for multipartite states in all dimensions. In addition we show that if an entangled state is full rank, then its CSS is unique. For the bipartite case of two qubits our formula reduce to the one given in Phys. Rev. A 78, 032310 (2008).

quant-ph↗
arXiv2010

On the minimum rank of a graph over finite fields

In this paper we deal with two aspects of the minimum rank of a simple undirected graph $G$ on $n$ vertices over a finite field $\FF_q$ with $q$ elements, which is denoted by $\mr(\FF_q,G)$. In the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on $n$ vertices over $\FF_2$ is $(1-\varepsilon_n)n$, were $\lim_{n\to\infty} \varepsilon_n=0$. In the second part of this paper we assume that $G$ contains a clique $K_k$ on $k$-vertices. We show that if $q$ is not a prime then $\mr(\FF_q,G)\le n-k+1$ for $4\le k\le n-1$ and $n\ge 5$. It is known that $\mr(\FF_q,G)\le 3$ for $k=n-2$, $n\ge 4$ and $q\ge 4$. We show that for $k=n-2$ and each $n\ge 10$ there exists a graph $G$ such that $\mr(\FF_3,G)>3$. For $k=n-3$, $n\ge 5$ and $q\ge 4$ we show that $\mr(\FF_q,G)\le 4$.

math.CO↗
arXiv2010

The pressure, densities and first order phase transitions associated with multidimensional SOFT

We study theoretical and computational properties of the pressure function for subshifts of finite type on the integer lattice $\Z^d$, multidimensional SOFT, which are called Potts models in mathematical physics. We show that the pressure is Lipschitz and convex. We use the properties of convex functions to show rigorously that the phase transition of the first order correspond exactly to the points where the pressure is not differentiable. We give computable upper and lower bounds for the pressure, which can be arbitrary close the values of the pressure given a sufficient computational power. We apply our numerical methods to confirm Baxter's heuristic computations for two dimensional monomer-dimer model, and to compute the pressure and the density entropy as functions of two variables for the two dimensional monomer-dimer model.

math-ph↗
arXiv2009

Graph isomorphism and volumes of convex bodies

We show that a nontrivial graph isomorphism problem of two undirected graphs, and more generally, the permutation similarity of two given $n\times n$ matrices, is equivalent to equalities of volumes of the induced three convex bounded polytopes intersected with a given sequence of balls, centered at the origin with radii $t_i\in (0,\sqrt{n-1})$, where $\{t_i\}$ is an increasing sequence converging to $\sqrt{n-1}$. These polytopes are characterized by $n^2$ inequalities in at most $n^2$ variables. The existence of fpras for computing volumes of convex bodies gives rise to a semi-frpas of order $O^*(n^{14})$ at most to find if given two undirected graphs are isomorphic.

cs.CC↗
arXiv2009

The 1-vertex transfer matrix and accurate estimation of channel capacity

The notion of a 1-vertex transfer matrix for multi-dimensional codes is introduced. It is shown that the capacity of such codes, or the topological entropy, can be expressed as the limit of the logarithm of spectral radii of 1-vertex transfer matrices. Storage and computations using the 1-vertex transfer matrix are much smaller than storage and computations needed for the standard transfer matrix. The method is applied to estimate the first 15 digits of the entropy of the 2-dimensional (0,1) run length limited channel. In order to compare the computational cost of the new method with the standard transfer matrix and have rigorous bounds to compare the estimates with a large scale computation of eigenvalues for the (0,1) run length limited channel in 2 and 3 dimensions have been carried out. This in turn leads to improvements on the best previous lower and upper bounds for that channel.

math-ph↗
arXiv2009

Additive invariants on quantum channels and applications to regularized minimum entropy

We introduce two additive invariants of output quantum channels. If the value of one these invariants is less than 1 then the logarithm of the inverse of its value is a positive lower bound for the regularized minimum entropy of an output quantum channel. We give a few examples in which one of these invariants is less than 1. We also study the special cases where the above both invariants are equal to 1.

quant-ph↗