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In quantum mechanics, the Heisenberg uncertainty relation presents an ultimate limit to the precision by which one can predict the outcome of position and momentum measurements on a particle. Heisenberg explicitly stated this relation for the prediction of hypothetical future measurements, and it does not describe the situation where knowledge is available about the system both earlier and later than the time of the measurement. We study what happens under such circumstances with an atomic ensemble containing $10^{11}$ $^{87}text{Rb}$ atoms, initiated nearly in the ground state in presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, $hat{x}_A$ and $hat{p}_A$ that satisfy $[hat{x}_A,hat{p}_A]=ihbar$. Quantum non-demolition measurements of $hat{p}_A$ before and of $hat{x}_A$ after time $t$ allow precise estimates of both observables at time $t$. The capability of assigning precise values to multiple observables and to observe their variation during physical processes may have implications in quantum state estimation and sensing.
We prove a double-inequality for the product of uncertainties for position and momentum of bound states for 1D quantum mechanical systems in the semiclassical limit.
Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenbergs error-disturbance relation. In contrast, we ha
Quantum uncertainty relations are formulated in terms of relative entropy between distributions of measurement outcomes and suitable reference distributions with maximum entropy. This type of entropic uncertainty relation can be applied directly to o
We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when th
A Heisenberg uncertainty relation is derived for spatially-gated electric and magnetic field fluctuations. The uncertainty increases for small gating sizes which implies that in confined spaces the quantum nature of the electromagnetic field must be