The common invariant subspace problem and Tarski’s theorem

Abstract

This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is elementary and based on the effective Nullstellensatz. The existence of a common invariant subspace of $A_{1},\dots,A_{s}$ of dimension $d$ can be expressed in the first-order language of fields, and hence, the constructive version of Tarski's theorem yields the criterion. In addition, some applications of this criterion in quantum information theory are discussed.