Defect of a Kronecker product of unitary matrices

TL;DR

This paper investigates the defect of Kronecker product unitary matrices, providing a method to compute and bound the defect, and demonstrating supermultiplicativity properties relevant for understanding matrix structure.

Contribution

It introduces a splitting technique for calculating the defect of Kronecker product unitaries and establishes a lower bound and supermultiplicativity of the defect.

Findings

A method to compute the defect of Kronecker product unitaries.

A lower bound on the generalized defect D(U).

Supermultiplicativity of the defect D(U) with respect to Kronecker subproducts.

Abstract

The generalized defect D(U) of a unitary NxN matrix U with no zero entries is the dimension of the real space of directions, moving into which from U we do not disturb the moduli |U_ij| as well as the Gram matrix U'*U in the first order. Then the defect d(U) is equal to D(U) - (2N-1), that is the generalized defect diminished by the dimension of the manifold {Dr*U*Dc : Dr,Dc unitary diagonal}. Calculation of d(U) involves calculating the dimension of the space in R^(N^2) spanned by a certain set of vectors associated with U. We split this space into a direct sum, assuming that U is a Kronecker product of unitary matrices, thus making it easier to perform calculations numerically. Basing on this, we give a lower bound on D(U) (equivalently d(U)), supposing it is achieved for most unitaries with a fixed Kronecker product structure. Also supermultiplicativity of D(U) with respect to Kronecker subproducts of U is shown.