Low rank and sparse dynamical maps and repeated entries in the process matrix

TL;DR

This paper demonstrates that low-rank and sparse dynamical maps have process matrices with only a few distinct entries, scaling as the square of the rank, in specific operator bases.

Contribution

It reveals that low-rank and sparse dynamical maps exhibit a structured process matrix with repeated entries, reducing complexity in characterization.

Findings

Process matrices have O(r^2) distinct entries for low-rank maps.

Repeated entries in the process matrix simplify analysis.

Applicable in bases of Pauli operators or SU(N) generators.

Abstract

We show that the process matrix in the basis of tensor products of Pauli operators or SU(N) generators representing low rank and sparse dynamical maps will have only a few distinct entries which goes as $O(r^2)$ ($r$ is the rank).