  We reveal the continuous phase transition in anyonic-PT symmetric systems,
contrasting with the discontinuous phase transition corresponding to the
discrete (anti-) PT symmetry. The continuous phase transition originates from
the continuity of anyonic-PT symmetry. We find there are three
information-dynamics patterns for anyonic-PT symmetric systems: damped
oscillations with an overall decrease (increase) and asymptotically stable
damped oscillations, which are three-fold degenerate and distorted using the
Hermitian quantum R\'enyi entropy or distinguishability. It is the
normalization of the non-unitary evolved density matrix causes the degeneracy
and distortion. We give a justification for non-Hermitian quantum R\'enyi
entropy being negative. By exploring the mathematics and physical meaning of
the negative entropy in open quantum systems, we connect the negative
non-Hermitian quantum R\'enyi entropy and negative quantum conditional entropy,
opening up a new journey to rigorously investigate the negative entropy in open
quantum systems.
