Observing a Changing Hilbert-Space Inner Product

Abstract

In quantum mechanics, physical states are represented by rays in Hilbert space $\mathcal{H}$, which is a vector space imbued by an inner product $⟨|⟩$, whose physical meaning arises as the overlap $⟨\varphi |\psi ⟩$ for $|\psi ⟩$ a pure state (description of preparation) and $⟨\varphi |$ 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.

12 pages, 4 figures, updated version includes an erratum