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Chin. Phys. B, 2022, Vol. 31(11): 110307    DOI: 10.1088/1674-1056/ac76b4
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Quantum speed limit for mixed states in a unitary system

Jie-Hui Huang(黄接辉)1,2,†, Li-Guo Qin(秦立国)1, Guang-Long Chen(陈光龙)1, Li-Yun Hu(胡利云)2,‡, and Fu-Yao Liu(刘福窑)1,§
1 School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China;
2 College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China
Abstract  Since the evolution of a mixed state in a unitary system is equivalent to the joint evolution of the eigenvectors contained in it, we could use the tool of instantaneous angular velocity for pure states to study the quantum speed limit (QSL) of a mixed state. We derive a lower bound for the evolution time of a mixed state to a target state in a unitary system, which automatically reduces to the quantum speed limit induced by the Fubini-Study metric for pure states. The computation of the QSL of a degenerate mixed state is more complicated than that of a non-degenerate mixed state, where we have to make a singular value decomposition (SVD) on the inner product between the two eigenvector matrices of the initial and target states. By combing these results, a lower bound for the evolution time of a general mixed state is presented. In order to compare the tightness among the lower bound proposed here and lower bounds reported in the references, two examples in a single-qubit system and in a single-qutrit system are studied analytically and numerically, respectively. All conclusions derived in this work are independent of the eigenvalues of the mixed state, which is in accord with the evolution properties of a quantum unitary system.
Keywords:  quantum speed limit      instantaneous angular velocity      singular value decomposition  
Received:  23 January 2022      Revised:  17 May 2022      Accepted manuscript online:  08 June 2022
PACS:  03.65.-w (Quantum mechanics)  
  03.65.Fd (Algebraic methods)  
  03.65.Ta (Foundations of quantum mechanics; measurement theory)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11664018, 12174247, and U2031145).
Corresponding Authors:  Jie-Hui Huang, Li-Yun Hu, Fu-Yao Liu     E-mail:;;

Cite this article: 

Jie-Hui Huang(黄接辉), Li-Guo Qin(秦立国), Guang-Long Chen(陈光龙), Li-Yun Hu(胡利云), and Fu-Yao Liu(刘福窑) Quantum speed limit for mixed states in a unitary system 2022 Chin. Phys. B 31 110307

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