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arXiv · 2101.00015

Observing a Changing Hilbert-Space Inner Product

Salini Karuvade·Abhijeet Alase·Barry C. Sanders
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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.

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Salini Karuvade, Abhijeet Alase, Barry C. Sanders. 2020-12-31. Observing a Changing Hilbert-Space Inner Product. https://doi.org/10.1103/physrevresearch.4.013016

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