Discriminating an Arbitrary Number of Pure Quantum States by the Combined $\mathcal{C}\mathcal{P}\mathcal{T}$ and Hermitian Measurements

If the system is known to be in one of two non-orthogonal quantum states, $\mathrm{\mid }{\psi }_1⟩$ or $\mathrm{\mid }{\psi }_2⟩$, $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics can discriminate them, $\mathit{\text{in}}\mathit{\text{principle}}$, by a single measurement. We extend this approach by combining $\mathcal{P}\mathcal{T}$-symmetric and Hermitian measurements and show that it’s possible to distinguish an arbitrary number of pure quantum states by an appropriate choice of the parameters of $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian.