Observing a Changing Hilbert-Space Inner Product

In quantum mechanics, physical states are represented by rays in Hilbert space, which is a vector space imbued by an inner product. For a given pair of pure states ψ, φ, physical meaning of the inner product between the two arises as the probability amplitude associated to observing the system in φ when the system was originally prepared in ψ. Current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement of the system. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum channel, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. Physically, we explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.