Observing a changing Hilbert-space inner product

In quantum mechanics, physical states are represented by rays in Hilbert space H , which is a vector space imbued by an inner product «|» , whose physical meaning arises as the overlap «ϕ |ψ » for |ψ » a pure state (description of preparation) and «ϕ | a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. 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 operation, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.