Adjoint master equation for multi-time correlators

The quantum regression theorem is a powerful tool for calculating the muli-time correlators of operators of open quantum systems which dynamics can be described in Markovian approximation. It enables to obtain the closed system of equation for the multi-time correlators. However, the scope of the quantum regression theorem is limited by a particular time order of the operators in multi-time correlators and does not include out-of-time-ordered correlators. In this work, we obtain an adjoint master equation for multi-time correlators that is applicable to out-of-time-ordered correlators. We show that this equation can be derived for various approaches to description of the dynamics of open quantum systems, such as the global or local approach. We show that the adjoint master equation for multi-time correlators is self-consistent. Namely, the final equation does not depend on how the operators are grouped inside the correlator, and it coincides with the quantum regression theorem for the particular time ordering of the operators.
Comment: 11 pages