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

TL;DR

This paper proves that the cost function in the Krotov algorithm for open quantum systems monotonically increases, providing a rigorous mathematical foundation for its convergence in quantum control tasks.

Contribution

It offers a detailed proof of the monotonic increase of the Krotov algorithm's cost function specifically for open quantum systems, expanding upon prior brief treatments.

Findings

Proof of monotonic increase in the cost function

Mathematical validation of Krotov algorithm convergence

Enhanced theoretical understanding of quantum control methods

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.