Switching Quantum Reference Frames for Quantum Measurement

Jianhao M. Yang

Qualcomm, San Diego, CA 92121, USA

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Physical observation is made relative to a reference frame. A reference frame is essentially a quantum system given the universal validity of quantum mechanics. Thus, a quantum system must be described relative to a quantum reference frame (QRF). Further requirements on QRF include using only relational observables and not assuming the existence of external reference frame. To address these requirements, two approaches are proposed in the literature. The first one is an operational approach (F. Giacomini, et al, Nat. Comm. 10:494, 2019) which focuses on the quantization of transformation between QRFs. The second approach attempts to derive the quantum transformation between QRFs from first principles (A. Vanrietvelde, et al, $\textit{Quantum}$ 4:225, 2020). Such first principle approach describes physical systems as symmetry induced constrained Hamiltonian systems. The Dirac quantization of such systems before removing redundancy is interpreted as perspective-neutral description. Then, a systematic redundancy reduction procedure is introduced to derive description from perspective of a QRF. The first principle approach recovers some of the results from the operational approach, but not yet include an important part of a quantum theory - the measurement theory. This paper is intended to bridge the gap. We show that the von Neumann quantum measurement theory can be embedded into the perspective-neutral framework. This allows us to successfully recover the results found in the operational approach, with the advantage that the transformation operator can be derived from the first principle. In addition, the formulation presented here reveals several interesting conceptual insights. For instance, the projection operation in measurement needs to be performed after redundancy reduction, and the projection operator must be transformed accordingly when switching QRFs. These results represent one step forward in understanding how quantum measurement should be formulated when the reference frame is also a quantum system.

The paper is a step forward in the investigation of quantum measurement with quantum reference frames (QRF). Utilizing the recently published first-principle approach to the formulation of quantum mechanics when switching QRFs, the paper addresses the following issue of quantum measurement: How should the same measurement process be described from the points of view of different QRFs? How do the descriptions relate to one another? In the context of the first-principle approach, the answer is this: the unitary evolution of the measurement process can be embedded in a perspective-neutral framework, but the measurement projection is perspectival, and thus shall be implemented after the QRF is specified.

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Cited by

[1] Pierre Martin-Dussaud, "Perspective on: Switching Quantum Reference Frames for Quantum Measurement", Quantum Views 4, 40 (2020).

[2] Jianhao M. Yang, "Quantum Mechanics from Relational Properties, Part III: Path Integral Implementation", arXiv:1807.01583.

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