  Recently, Zheng constructs a quantum-control-assisted multipartite
variance-based uncertainty relation, which successfully extends the conditional
uncertainty relation to the multipartite case [Annalen der physik, 533, 2100014
(2021)]. We here investigate the dynamics of the new uncertainty relation in
the Heisenberg system with the Dzyaloshinski-Moriya interaction. It is found
that, different from entanglement, the mixedness of the system has an
interesting single-valued relationship with the tightness and lower bound of
the uncertainty relation. This single-valued relationship indicates that the
tightness and lower bound of the uncertainty relation can be written as the
functional form of the mixedness. Moreover, the single-valued relationship with
the mixedness is the common nature of conditional uncertainty relations, and
has no relationship with the form of the uncertainty relations. Also, the
comparison between the new conditional variance-based uncertainty relation and
the existing entropic one has been made.
