$\mathcal{PT}$-Symmetric Quantum Discrimination of Three States

If the system is known to be in one of two non-orthogonal quantum states, $|\psi_1\rangle$ or $|\psi_2\rangle$, it is not possible to discriminate them by a single measurement due to the unitarity constraint. In a regular Hermitian quantum mechanics, the successful discrimination is possible to perform with the probability $p < 1$, while in $\mathcal{PT}$-symmetric quantum mechanics a \textit{simulated single-measurement} quantum state discrimination with the success rate $p$ can be done. We extend the $\mathcal{PT}$-symmetric quantum state discrimination approach for the case of three pure quantum states, $|\psi_1\rangle$, $|\psi_2\rangle$ and $|\psi_3\rangle$ without any additional restrictions on the geometry and symmetry possession of these states. We discuss the relation of our approach with the recent implementation of $\mathcal{PT}$ symmetry on the IBM quantum processor.