Excess entropy production in quantum systems

For open systems described by the quantum master equation (QME), we investigate the excess entropy production under quasistatic operations between nonequilibrium steady states[1]. The average entropy production is composed of the time integral of the instantaneous steady entropy production rate and the excess entropy production. We propose to define average entropy production rate using the average energy and particle currents, which are calculated by using the full counting statistics with QME. The excess entropy production is given by a line integral in the control parameter space and its integrand is called the Berry–Sinitsyn–Nemenman (BSN) vector. In the weakly nonequilibrium regime, we devied the BSN vector with is described with the density operator of the instantaneous steady state of the QME and the density operator given by the QME with reversing the sign of the Lamb shift term. If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, the excess entropy production approximately reduces to the difference between the von Neumann entropies of the system.
[1] S. Nakajima and Y. Tokura, J. Stat. Phys. 169: 902. Sep. (2017).