  Non-Hermitian systems with parity-time reversal ($\mathcal{PT}$) or
anti-$\mathcal{PT}$ symmetry have attracted a wide range of interest owing to
their unique characteristics and counterintuitive phenomena. One of the most
extraordinary features is the presence of an exception point (EP), across which
a phase transition with spontaneously broken $\mathcal{PT}$ symmetry takes
place. We implement a Floquet Hamiltonian of a single qubit with
anti-$\mathcal{PT}$ symmetry by periodically driving a dissipative quantum
system of a single trapped ion. With stroboscopic emission and quantum state
tomography, we obtain the time evolution of density matrix for an arbitrary
initial state, and directly demonstrate information retrieval, eigenstates
coalescence, and topological energy spectra as unique features of non-Hermitian
systems.
