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We investigate the influence of environment noise on the control cost and thequantum speed limit time (QSLT) in the process of almost exact statetransmission (AEST) through a spin chain under pulse control. The chain isimmersed in its surrounding non-Markovian, finite temperature heat baths. Wefind that AEST can be realized in weak system-bath coupling, low temperature,and strong non-Markovian baths under effective external control.Correspondingly, the control cost and QSLT increases with increasing bathtemperature and coupling strength. It is noticeable that non-Markovianity fromthe baths can be helpful to reduce the control cost and shorten the QSLT.Furthermore, we find that there exists a trade-off between the control cost andtransmission fidelity and higher fidelity requires higher cost. In addition,the minimum control cost has been found to obtain certain transmissionfidelity.

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