Completely Reducible maps in Quantum Information Theory

TL;DR

This paper explores conditions under which associated self-adjoint maps in quantum information are completely reducible, providing new proofs and characterizations for PPT, SPC, and realignment-invariant matrices, with implications for quantum state decompositions.

Contribution

It establishes that certain classes of matrices in quantum information theory have completely reducible associated maps and characterizes these classes using group actions and invariance properties.

Findings

Associated self-adjoint maps are completely reducible for PPT, SPC, and realignment-invariant matrices.

Every realignment-invariant matrix in M_2⊗M_2 is PPT, with a counterexample for higher dimensions.

The paper provides new proofs for known theorems and characterizes matrix classes with reducible associated maps.

Abstract

In order to compute the Schmidt decomposition of $A\in M_k\otimes M_m$, 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 (SPC) 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 these theorems for matrices invariant under realignment using theorems of Perron-Frobenius theory. We also provide a new proof of the fact that if $\mathbb{C}^{k}$ contains $k$ mutually unbiased bases then $\mathbb{C}^{k}$ contains $k+1$. We search for other types of matrices that could have the same property. We consider a group of linear transformations acting on $M_k\otimes M_k$, which contains the partial transpositions and the realignment map. For each element of this group, we consider the set of matrices in $M_k\otimes M_k\simeq M_{k^2}$ that are positive and remain positive, or invariant, under the action of this element. 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 $M_2\otimes M_2$ and we present a counterexample in $M_k\otimes M_k$, $k\geq 3$.