The exact classical boundary

A diagonal Hamiltonian also preserves this boundary. The theorem does not include arbitrary coherent interventions or phase-sensitive readouts. Such operations can leave the specified classical observation class.

For nonzero Hamiltonian,

so coherence can influence populations. That fact motivates, but does not alone prove, a task improvement.

Quantum notation, diagonal state variables, and dissipator symbols are therefore not by themselves a quantum resource certificate.