Primitive quantum channel

A finite-dimensional quantum channel is primitive when some power maps every nonzero positive semidefinite operator to a positive-definite operator. Repeated application then forgets the initial state and approaches a unique stationary density operator with full rank.

An equivalent spectral criterion requires a full-rank stationary state, a simple eigenvalue one and no other eigenvalue on the unit circle. The full-rank condition is essential: an amplitude damping channel can have a unique stationary pure ground state and all other eigenvalues inside the unit circle without being primitive. A state starting in that ground state never becomes positive definite. Primitivity is therefore stronger than merely having a unique attractive stationary state, and its use in relaxation arguments must specify the finite-dimensional setting.