Positivity for Matrix Systems: A Case Study from Quantum Mechanics

We discuss an example from quantum physics of “positive system” in which the state (a density operator) is a square matrix constrained to be positive semidefinite (plus Hermitian and of unit trace). The positivity constraint is captured by the notion of complete positivity of the corresponding flow. The infinitesimal generators of all possible admissible ODEs can be characterized explicitly in terms of cones of matrices. Correspondingly, it is possible to determine all linear timevarying systems and bilinear control systems that preserve positivity of the state space.