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Uncertainty relation for the position of an electron in a uniform magnetic field from quantum estimation theory

We investigate the uncertainty relation for estimating the position of one electron in a uniform magnetic field in the framework of the quantum estimation theory. Two kinds of momenta, canonical one and mechanical one, are used to generate a shift in the position of the electron. We first consider pure state models whose wave function is in the ground state with zero angular momentum. The model generated by the two-commuting canonical momenta becomes the quasi-classical model, in which the symmetric logarithmic derivative quantum Cramér-Rao bound is achievable. The model generated by the two non-commuting mechanical momenta, on the other hand, turns out to be a Gaussian model, where the generalized right logarithmic derivative quantum Cramér-Rao bound is achievable. We next consider mixed-state models by taking into account the effects of thermal noise. The model with the canonical momenta now becomes genuine quantum mechanical, although its generators commute with each other. The derived uncertainty relationship is in general determined by two different quantum Cramér-Rao bounds in a non-trivial manner. The model with the mechanical momenta is identified with the well-known Gaussian shift model, and the uncertainty relation is governed by the right logarithmic derivative Cramér-Rao bound.