Unambiguous Quantum State Discrimination in a PT $\mathcal {PT}$ -Symmetric System of a Single Trapped Ion

An unambiguous quantum state discrimination of two qubit states is experimentally demonstrated under a non-Hermitian Hamiltonian with parity-time-reversal ( $\mathrm{PT}$ ) symmetry in a single trapped ⁴⁰Ca⁺ ion. It is shown that any two non-orthogonal states can become orthogonal subjected to time evolution of a $\mathrm{PT}$ -symmetric Hamiltonian in both the $\mathrm{PT}$ -symmetry preserving and broken regimes, thus can be discriminated unambiguously. For a given pair of candidate states, it is shown that the parameters of the Hamiltonian must be confined in a proper range, within which there exists an optimal choice to realize quantum brachistochrone for the fastest orthogonalization. Besides, a clear geometric picture and some analytic results are provided to understand the main conclusions. This work shows a promising application of non-Hermitian physics in quantum information processing.