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

TL;DR

This paper develops a computable criterion for common eigenvectors and invariant subspaces of matrices, extending Shemesh's result, with applications to quantum information theory involving irreducible completely positive superoperators.

Contribution

It generalizes Shemesh's criterion to multiple matrices and provides conditions for common invariant subspaces, with implications for quantum information theory.

Findings

A new criterion for common eigenvectors of multiple matrices.

Necessary and sufficient conditions for common invariant subspaces.

Most matrices with multiple eigenvalues are negligible in measure, making the criterion broadly applicable.

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 $d$, $2\leq d<n$, 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.