Control cost and quantum speed limit time in controlled almost exact state transmission in open systems

Abstract

We investigate the influence of the environment noise on the control cost and quantum speed limit time (QSLT) in the process of quantum state transmission through a spin chain under pulse control. The chain is immersed in its surrounding non-Markovian, finite temperature heat baths. We find that almost exact state transmission (AEST) can be realized in weak system-bath coupling, low temperature, and strong non-Markovian baths under effective external control. Correspondingly, the control cost decreases with decreasing bath temperature, system-bath coupling strength. Non-Markovianity from the baths can also help to reduce the cost. There exists a trade-off between the control cost and transmission fidelity. Higher fidelity requires higher cost. In addition, the minimum cost has been found to obtain AEST. Furthermore, we discuss the QSLT during the process of AEST. It is noticeable that non-Markovianity from the baths plays a very important role. More non-Markvoian baths correspond to shorter QSLT for weak system-bath coupling and low temperature. However, in the strong coupling and high temperature regime, more Markovian baths correspond to shorter QSLT.