Lower Bounds for Matrix Factorization

Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of o...

Lower Bounds for Matrix Product

We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two n n matrices over finite fields. In particular we obtain the following results: 1. We show that the number of product gates in any bilinear (or quadratic) circuit that computes the product of two n n matrices over GF (2) is at least 3n2 o(n2). 2. We show that the number of pro...

Some Lower Bounds for Estrada Index

Factorization law for two lower bounds of concurrence

Entanglement, one of the important features of quantum systems, which does not exist classically, has been known as a key resource for some quantum computation and information processes. But the entanglement of a system changes due to its unavoidable interactions with environment. To study the entanglement changes, one needs to make use of an entanglement measure in order to specify the entangl...

Some inequalities involving lower bounds of operators on weighted sequence spaces by a matrix norm

Let A = (an;k)n;k1 and B = (bn;k)n;k1 be two non-negative ma-trices. Denote by Lv;p;q;B(A), the supremum of those L, satisfying the followinginequality:k Ax kv;B(q) L k x kv;B(p);where x 0 and x 2 lp(v;B) and also v = (vn)1n=1 is an increasing, non-negativesequence of real numbers. In this paper, we obtain a Hardy-type formula forLv;p;q;B(H), where H is the Hausdor matrix and 0 < q p 1. Also...