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The textbook uncertainty relation due to Kennard, Weyl and Robertson is a "preparation uncertainty relation", which constrains how sharply the position and momentum distributions of a fixed state can be concentrated. It has nothing to say about error -disturbance tradeoffs for setups like Heisenberg's gamma-ray microscope from 1927. Formulated by Heisenberg only as a heuristic principle, this tradeoff links the accuracy of an approximate position measurement to the possibility to retrieve the momentum after the measurement. We generalize this problem to a simpler one, namely the quality of optimal joint measurements of position and momentum. It is then only required that a device provides a position value and a momentum value in each shot, like the microscope and the momentum retrieval in the error/disturbance case. We will show quantitatively that the distribution of the position outputs must in general differ from that of an ideal position measurement, and similarly for momentum, where the respective deviations satisfy a relation looking formally like the textbook preparation uncertainty relation. Our setup is conceptually straightforward and covers arbitrary pairs or tuples of observables. It turns out that the equality of the preparation and measurement uncertainty bounds is due to the high symmetry of canonical pairs of observables linked via Fourier transform. For generic pairs of observables the measurement uncertainty is larger than the preparation uncertainty.

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Reinhard Werner, Leibniz-Universität, Hannover
Seminar room 0.03, ETP
Contact: David Gross