Correlation evolution and monogamy of two geometric quantum discords in multipartite systems

We explore two different geometric quantum discords defined respectively via the trace norm (GQD-1) and Hilbert-Schmidt norm (GQD-2) in multipartite systems. A rigorous hierarchy relation is revealed for the two GQDs in a class of symmetric two-qubit $X$-shape states. For multiqubit pure states, it is found that both GQDs are related to the entanglement concurrence, with the hierarchy relation being saturated. Furthermore, we look into a four-partite dynamical system consisting of two cavities interacting with independent reservoirs. It is found that the GQD-2 can exhibit various sudden change behaviours, while the GQD-1 only evolves asymptotically, with the two GQDs exhibiting different monogamous properties.