Observing a Changing Hilbert-Space Inner Product

• URL: http://arxiv.org/abs/2101.00015v3
• Date: Thu, 30 Nov 2023 09:58:53 GMT
• Title: Observing a Changing Hilbert-Space Inner Product
• Abstract summary: Current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We show that this change is described by a quantum channel, which is tomographically observable. 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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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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