PT-symmetric quantum state discrimination

Suppose that a system is known to be in one of two quantum states, $\mathrm{\mid }{\psi }_1>$ or $\mathrm{\mid }{\psi }_2>$. If these states are not orthogonal, then in conventional quantum mechanics it is impossible with one measurement to determine with certainty which state the system is in. However, because a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states, it is always possible to choose this inner product so that the two states $\mathrm{\mid }{\psi }_1>$ and $\mathrm{\mid }{\psi }_2>$ are orthogonal. Thus, quantum state discrimination can, in principle, be achieved with a single measurement.