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The uncertainty relation for joint measurement of position and momentum
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abstract
We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form $(\Delta P)(\Delta Q)\geq C\hbar$, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision $\Delta Q$ of the position measurement, and the perturbation $\Delta P$ of the conjugate variable introduced by such a measurement. We also determine the best constant $C$, which is attained for a unique phase space covariant measurement.
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Cited by 1 Pith paper
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Uncertainty Relations Relative to Phase-Space Quantum Reference Frames
Frame-conditioned position and momentum are jointly measurable under a covariant phase-space quantum reference frame, yielding new uncertainty bounds that reduce to standard Heisenberg bounds in a classical frame limit.
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