Recovery map stability for the data processing inequality

Let be a finite dimensional von Neumann algebra and a von Neumann subalgebra of it. For states and on , let and be the corresponding states induced on . The data processing inequality implies that where is the relative entropy. Petz proved that there is equality if and only if , where is the Petz recovery map. We prove a quantitative version of Petz's theorem. In it simplest form, our bound is where is the relative modular operator. Since , this yields a bound that is independent of . We also prove an analogous result with a more complicated constant in which the roles of and are interchanged on the right. Quantum information theoretic inequalities are usually much harder to prove, or differ from, their classical counterparts because classical proofs often rely on conditioning argument that do not carry over to the quantum setting. In particular, quantum conditional expectations rarely preserve expectations-something that always happens in the classical setting. We also prove a simple theorem characterizing states and subalgebras for which conditional expectations do preserve expectation with respect to , illuminating the quantum obstacle to the existence of nicely behaved conditional expectations and the origins the Petz recovery map.