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Information retrieval and eigenstates coalescence in a non-Hermitian quantum system with anti-$\mathcal{PT}$ symmetry

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.