  The $\mathcal{PT}$-symmetric non-Hermitian systems have been widely studied
and explored both in theory and in experiment these years due to various
interesting features. In this work, we focus on the dynamical features of a
triple-qubit system, one of which evolves under local $\mathcal{PT}$-symmetric
Hamiltonian. A new kind of abnormal dynamic pattern in the entropy evolution
process is identified, which presents a parameter-dependent stable state,
determined by the non-Hermiticity of Hamiltonian in the broken phase of
$\mathcal{PT}$-symmetry. The entanglement and mutual information of a two-body
subsystem can increase beyond the initial values, which do not exist in the
Hermitian and two-qubit $\mathcal{PT}$-symmetric systems. Moreover, an
experimental demonstration of the stable states in non-Hermitian system with
non-zero entropy and entanglement is realized on a four-qubit quantum simulator
with nuclear spins. Our work reveals the distinctive dynamic features in the
triple-qubit $\mathcal{PT}$-symmetric system and paves the way for practical
quantum simulation of multi-party non-Hermitian system on quantum computers.
