Full-parameter-space quantum simulation of high-fidelity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="script">PT</mml:mi> </mml:math> -symmetric dynamics in a trapped ion

In recent years, non-Hermitian parity-time ($\mathcal{P}\mathcal{T}$)-symmetric quantum systems have gained significant attention due to their unique physical properties, with experimental realizations demonstrated across diverse platforms. Two popular approaches have been used to implement $\mathcal{P}\mathcal{T}$-symmetric dynamics: one removes quantum jumps from dissipative natural atoms (e.g., cold atoms, trapped ions) or artificial atoms (e.g., superconducting qubits) through postselection whose experimental fidelity is severely limited by unavoidable population backflow effects; the other embeds the desired non-Hermitian Hamiltonian into a larger Hermitian system, which achieves higher fidelity but typically enables simulating $\mathcal{P}\mathcal{T}$-symmetric dynamics only within restricted parameter regimes. This paper proposes and experimentally demonstrates a trapped-ion-based scheme for digital simulation of $\mathcal{P}\mathcal{T}$-symmetric quantum systems with high fidelity across the full parameter space. By leveraging high-fidelity logic gates and postselection on a four-level single ion to synthesize the target $\mathcal{P}\mathcal{T}$-symmetric dynamics, this approach enables accurate characterization of $\mathcal{P}\mathcal{T}$-symmetric properties, especially the exotic features in the vicinity of the exceptional point (EP). Distinct dynamical properties in the $\mathcal{P}\mathcal{T}$-symmetric unbroken phase (PTS), $\mathcal{P}\mathcal{T}$-symmetric broken phase (PTB), and at the EP are experimentally demonstrated with the data agreeing well with theoretical predictions. This high-fidelity simulation scheme also enables precise measurement of quantum Fisher information (QFI) near the EP, essential for evaluating non-Hermitian quantum sensing performance. It is shown that, after accounting for the low postselection probability, QFI divergence near the EP vanishes, providing no more information than their Hermitian counterparts, which echoes with recent theoretical findings. Our scheme establishes a paradigm for full-parameter-space precise simulation of non-Hermitian effects in quantum systems, and is applicable to various other platforms.