Completely Reducible Maps in Quantum Information Theory

In order to compute the Schmidt decomposition of A ∈ M
_k
⊗ M
_m
, we must consider an associated self-adjoint map. Here, we show that if A is positive under partial transposition (PPT) or symmetric with positive coefficients (SPCs) or invariant under realignment, then its associated self-adjoint map is completely reducible. We give applications of this fact in quantum information theory. We recover some theorems (recently proved for PPT and SPC matrices), and we prove them for matrices invariant under realignment using theorems of the Perron-Frobenius theory. We also provide a new proof of the fact that if C
^k
contains k mutually unbiased bases, then there exists another orthonormal basis which is mutually unbiased with these k bases. We study other types of matrices that could have the same property. We consider a collection of linear transformations acting on M
_k
⊗ M
_k
, which contains the partial transpositions and the realignment map. For each linear transformation, we consider the set of matrices in M
_k
⊗ M
_k
≃ M(k
²
) that are positive and remain positive, or invariant, under the action of this linear transformation. Within this family of sets, we have the set of PPT matrices, the set of SPC matrices and the set of matrices invariant under realignment. We show that these three sets are the only sets of this family, such that the associated self-adjoint map of each matrix is completely reducible. We also show that every matrix invariant under realignment is PPT in M
₂
⊗ M
₂
and we present a counterexample in M
_k
⊗ M
_k
and k ≥ 3.