Abstract
The conservation law for first-order coherence and mutual correlation of a bipartite qubit state is first proposed by Svozilík et al. [Phys. Rev. Lett. 115, 220501 (2015)], and their theories laid the foundation for the study of coherence migration under unitary transformations. In this paper, we generalize the framework of first-order coherence and mutual correlation to an arbitrary $(m \otimes n)$-dimensional bipartite composite state by introducing an extended Bloch decomposition form of the state. We also generalize two kinds of unitary operators in high-dimensional systems, which can bring about coherence migration and help to obtain the maximum or minimum first-order coherence. Meanwhile, coherence migration in open quantum systems are investigated. We take depolarizing channels as examples and establish that the reduced first-order coherence of the principal system over time is completely transformed into mutual correlation of the $(2 \otimes 4)$-dimensional system-environment bipartite composite state. It is expected that our results may provide a valuable idea or method for controlling the quantum resource such as coherence and quantum correlations.
High-dimensional unitary transformations and boson sampling on temporal modes using dispersive optics
