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

Abstract

If the system is known to be in one of two non-orthogonal quantum states, $|{\psi }_1⟩$ or $|{\psi }_2⟩$, 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 $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics a \textit{simulated single-measurement} quantum state discrimination with the success rate $p$ can be done. We extend the $\mathcal{P}\mathcal{T}$-symmetric quantum state discrimination approach for the case of three pure quantum states, $|{\psi }_1⟩$, $|{\psi }_2⟩$ and $|{\psi }_3⟩$ 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 $\mathcal{P}\mathcal{T}$ symmetry on the IBM quantum processor.