Observing a changing Hilbert-space inner product

In quantum mechanics, physical states are represented by rays in Hilbert space $\mathcal{H}$, which is a vector imbued an inner product $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$, whose meaning arises as the overlap $\ensuremath{\langle}\ensuremath{\phi}|\ensuremath{\psi}\ensuremath{\rangle}$ for $|\ensuremath{\psi}\ensuremath{\rangle}$ pure state (description of preparation) and $\ensuremath{\langle}\ensuremath{\phi}|$ projective measurement. However, current theory does not formally address consequences changing during interval between preparation We establish theoretical framework such product, we show consistent with standard mechanics. Furthermore, that this change described operation, tomographically observable, elucidate how our result strongly related to exploding topic PT-symmetric explain realize experimentally qubit terms qutrit protocol unitary channel.