Oscillation probabilities for a $\rho$$\tau$-symmetric non-Hermitian two-state system

There is growing interest in viable quantum theories with$\rho$$\tau$-symmetric non-Hermitian Hamiltonians, but a formulation of transition matrix elements consistent with positivity and perturbative unitarity has so far proved elusive. This Letter provides such a formulation, which relies crucially on the ability to span the state space in such a way that the interaction and energy eigenstates are orthonormal with respect to the same positive-definite inner product. We mention possible applications to the oscillations of mesons and neutrinos.