Flat extension technique for moment matrices of positive linear functionals over mixed polynomials and an application in quantum information

Abstract

A mixed polynomial, or a Hermitian polynomial, is a (multivariate) complex polynomial with monomials in complex variables and their conjugates. This paper deals with two types of mixed polynomials: sums of squared magnitudes of mixed polynomials (SOS polynomials for short) and those of usual complex polynomials (shortly, SQN polynomials). It is obvious that SQN polynomials are SOS, but generally, the converse is invalid. Both SOS and SQN polynomials are always mixed ones. This paper aims to provide sufficient and necessary conditions for a mixed polynomial to be SOS or SQN via moment matrices and the polynomial degree. We then give a sufficient condition for a SOS polynomial to be SQN, based on its degree. To this end, we apply the flat extension theory to the moment matrices of SOS and SQN polynomials and consider some optimization problems over positive linear functionals defined on the \(*\)-vector space of these two polynomial types. Consequently, we also give a degree characteristic of polynomials in the radical of the SQN set, which is one of the interesting problems posed by D’Angelo (Adv Math 226:4607–4637, 2011). An application in quantum information is also discussed: We introduce a class of quantum matrices whose degree-four SQN polynomials simultaneously satisfy the nonnegative, SQN and SOS properties.