The rank of the matrix multiplication operator for n×n matrices is one of the most studied quantities in algebraic complexity theory. I prove that the rank is at least 3n2−o(n2). More precisely, for any integer p ≤ n− 1 the rank is at least (3− 1 p+1 )n2 − (1 + 2p 2p p−1 ) )n. The previous lower bound, due to Bläser, was 5 2 n2−3n (the case p = 1). The new bounds improve Bläser’s bound for all ...

In this paper we investigate partial spreads of H(2n− 1, q2) through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic sem...

It is well known that the set of separable pure states is measure 0 in the set of pure states. Herein we extend this fact and show that the set of rank r separable states is measure 0 in the set of rank r states provided r is not maximal rank. Recently quite a few authors have looked at low rank separable and entangled states. (See [1] and the references therein and [2].) Therefore it makes sen...

We define the Augmentation property for binary matrices with respect to different rank functions. A matrix A has the Augmentation property for a given rank function, if for any subset of column vectors x1, ..., xt for for which the rank of A does not increase when augmented separately with each of the vectors xi, 1 ≤ i ≤ t, it also holds that the rank does not increase when augmenting A with al...