Proof of monotonic increase in the cost function for Krotov algorithm for open quantum systems

Abstract

A great number of quantum control papers have used one of the variants of the monotonically convergent variational control algorithm of Krotov (as described in Maday and Turinici (2003), Tannor et al. (1992), Zhu and Rabitz (1998), etc). The paper "Speeding up Thermalisation via Open Quantum System Variational Optimisation" by N, Suri, et al. [EPJST 227, 203 -216 (2018), arXiv:1711.08776] provides us a way of carrying out Krotov algorithm for open quantum systems. We shall prove the Theorem 1 of the paper, greatly expanding upon the brief treatment given in appendix 1 of the same.