Abstract
“Protective measurement” refers to two related schemes for finding the expectation value of an observable without disturbing the state of a quantum system, given a single copy of the system that is subject to a “protecting” operation. There have been several claims that these schemes support interpreting the quantum state as an objective property of a single quantum system. Here we provide three counter-arguments, each of which we present in two versions tailored to the two different schemes. Our first argument shows that the same resources used in protective measurement can be used to reconstruct the quantum state in a different way via process tomography. Our second argument is based on exact analyses of special cases of protective measurement, and our final argument is to construct explicit “\(\psi \)-epistemic” toy models for protective measurement, which strongly suggest that protective measurement does not imply the reality of the quantum state. The common theme of the three arguments is that almost all of the information comes from the “protection” operation rather than the quantum state of the system, and hence the schemes have no implications for the reality of the quantum state.
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Notes
A similar argument was made by Rovelli [37].
We dare not ask how much money Alice has to pay Bob for this protection racket.
Any nondegenerate eigenstate with finite gaps to the neighbouring states would work just as well, but we use the ground state here for simplicity.
One might be concerned that the discontinuous change from \(\hat{H} = \hat{H}_\mathrm{S}\) to \(\hat{H} = \hat{H}_\mathrm{S} + g \hat{A} \otimes \hat{P}\) at \(t=0\) and back again at \(t = 1/g\) violates the assumptions of the adiabatic theorem. However, we can instead use the measurement interaction \(\hat{H}_\mathrm{I} = g(t) \hat{A} \otimes \hat{P}\) where g(t) is a smoothly varying function with \(\int _{t=0}^{t=T} g(t) \, \mathrm {d} t = 1\) and where \(g(t) = 0\) for \(t<0\) and \(t > T\).
This is not completely straightforward as Alice only knows \(\mathcal {C}\) as a linear map and not the specific decomposition in terms of the projectors given in Eq. (1). However, the fixed point set of \(\mathcal {C}\) is the set of operators that are diagonal in the basis, and there are several methods for determining the fixed point set of a completely-positive trace-preserving map, e.g. [48].
If an arbitrary nondegenerate eigenstate is used instead of the ground state, Alice must in addition measure \(\hat{H}_\mathrm{S}\) on the system to determine with certainty.
If we also take the limit \(\sigma \rightarrow 0\), so that \(\Phi \) is a Dirac delta, then the whole procedure amounts to a projective measurement of the observable analogous to equation 14 of [39].
Formally, if we specify an initial state of the pointer and then cast a final measurement of \(\hat{Q}\) as a POVM on the system, all of the POVM will be proportional to the identity.
You may alternatively call these “classical” models or “hidden variable theories”, depending on your personal terminology preferences.
See [55] for an interpretation of quantum state tomography compatible with the \(\psi \)-epistemic position.
We could easily include z-measurements as well, but having two nonorthogonal states is sufficient for determining whether protective measurement entails the reality of the quantum state.
That is, without any disturbance to (q, p). Of course our state of knowledge about the system would change, but nobody trying to learn about a system should want ”protection” from ”disturbance” to their knowledge of the system!
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Acknowledgements
MP is grateful to Aharon Brodutch and Shan Gao for discussions, in particular to Shan for correcting MP’s initial misunderstanding of the Zeno scheme. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. CF was supported by NSF Grant No. PHY-1212445, the Canadian Government through the NSERC PDF program, the IARPA MQCO program, the ARC via EQuS Project Number CE11001013, and by the US Army Research Office Grant Numbers W911NF-14-1-0098 and W911NF-14-1-0103. ML is supported by the Foundational Questions Institute (FQXi). We would like to thank Paul Merriam for a careful proof reading.
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Combes, J., Ferrie, C., Leifer, M.S. et al. Why protective measurement does not establish the reality of the quantum state. Quantum Stud.: Math. Found. 5, 189–211 (2018). https://doi.org/10.1007/s40509-017-0111-4
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DOI: https://doi.org/10.1007/s40509-017-0111-4