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Non-Hermitian systems with parity-time reversal ($\mathcal{PT}$) oranti-$\mathcal{PT}$ symmetry have attracted a wide range of interest owing totheir unique characteristics and counterintuitive phenomena. One of the mostextraordinary features is the presence of an exception point (EP), across whicha phase transition with spontaneously broken $\mathcal{PT}$ symmetry takesplace. We implement a Floquet Hamiltonian of a single qubit withanti-$\mathcal{PT}$ symmetry by periodically driving a dissipative quantumsystem of a single trapped ion. With stroboscopic emission and quantum statetomography, we obtain the time evolution of density matrix for an arbitraryinitial state, and directly demonstrate information retrieval, eigenstatescoalescence, and topological energy spectra as unique features of non-Hermitiansystems.

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