Second-order generators with quadratic Leibniz defect on full matrix algebras: a self-contained study of pure dephasing with Hamiltonian drift

This is a self-contained account, with complete proofs, of linear maps on the full matrix algebra whose Leibniz defect equals the product of the commutators with a fixed self-adjoint matrix H. Such maps are exactly one half of the double commutator with H plus a commutator with a second matrix K. For skew-adjoint K they generate unital, trace-preserving quantum Markov semigroups with a single self-adjoint Lindblad operator. Topics: normal form, reconstruction of H from the defect, and classification up to automorphisms; dynamics, duality and the Lindblad dissipation inequality; fixed points, an exact relaxation criterion and ergodicity without commutativity assumptions; the commutator energy, its Dirichlet-form property, and seven conditions equivalent to H and K commuting; a sharp complete modified logarithmic Sobolev inequality for pure dephasing with arbitrary spectrum; an exactly solved qubit with non-commuting drift; several self-adjoint Lindblad operators. Most of the material is known or elementary, and the text says so. The sharp entropy inequality and a finite-time optimal control result are taken from two companion notes by the author. Status: not reviewed by a human referee. The role of AI tools is described in the text.