Observing a Changing Hilbert-Space Inner Product

TL;DR

This paper develops a theoretical framework for understanding how a changing inner product in Hilbert space affects quantum states and measurements, linking it to quantum channels and PT-symmetric quantum mechanics.

Contribution

It introduces a formal model for a dynamic inner product in quantum mechanics, showing its consistency and experimental realizability via a qutrit protocol.

Findings

Changing inner products can be modeled as quantum channels.

The framework is experimentally realizable with a qutrit system.

The approach connects to PT-symmetric quantum mechanics.

Abstract

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.