The common invariant subspace problem and Tarski’s theorem

This article presents a computable criterion for the existence of a common invariant subspace of n×n complex matrices A1, . . . , As of a fixed dimension 1 ≤ d ≤ 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 φ of the language of fields, a quantifier-free formula φ′ such that φ↔φ′ in ACF is given explicitly. The construction of φ′ is elementary and based on the effective Nullstellensatz. The existence of a common invariant subspace of A1, . . . , As 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.