  In quantum mechanics, physical states are represented by rays in Hilbert
space $\mathscr H$, which is a vector space imbued by an inner product
$\langle\,|\,\rangle$, whose physical meaning arises as the overlap
$\langle\phi|\psi\rangle$ for $|\psi\rangle$ a pure state (description of
preparation) and $\langle\phi|$ 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 channel, 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.
