Generalized Shemesh criterion, common invariant subspaces and irreducible completely positive superoperators

Abstract

Assume that $A_1,...,A_s$ are complex $n\times n$ matrices. We give a computable criterion for existence of a common eigenvector of $A_i$ which generalize the result of D. Shemesh established for two matrices. We use this criterion to prove some necessary and sufficient condition for $A_i$ to have a common invariant subspace of dimension $d$, $2\le d<n$, if every $A_i$ has pairwise different eigenvalues. Finally, we observe that the set of all matrices having multiple eigevalues has Lebesgue measure 0 and thus the condition is sufficient in practical applications. Being motivated by quantum information theory, we give a flavour of such applications for irreducible completely positive superoperators.