Quantum information technologies require careful control for generating and preserving a desired target quantum state. The biggest practical obstacle is, of course, decoherence. Therefore, the reachability analysis, which in our scenario aims to estimate the distance between the controlled state under decoherence and the target state, is of great importance to evaluate the realistic performance of those technologies. This paper presents a lower bound of the fidelity-based distance for a general open Markovian quantum system driven by the decoherence process and several types of control including feedback. The lower bound is straightforward to calculate and can be used as a guide for choosing the target state, as demonstrated in some examples. Moreover, the lower bound is applied to derive a theoretical limit in some quantum metrology problems based on a large-size atomic ensemble under control and decoherence.
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