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Uncertainty relations for general phase spaces
Get PDFWe describe a setup for obtaining uncertainty relations for arbitrary pairs
of observables related by Fourier transform. The physical examples discussed
here are standard position and momentum, number and angle, finite qudit
systems, and strings of qubits for quantum information applications. The
uncertainty relations allow an arbitrary choice of metric for the distance of
outcomes, and the choice of an exponent distinguishing e.g., absolute or root
mean square deviations. The emphasis of the article is on developing a unified
treatment, in which one observable takes values in an arbitrary locally compact
abelian group and the other in the dual group. In all cases the phase space
symmetry implies the equality of measurement uncertainty bounds and preparation
uncertainty bounds, and there is a straightforward method for determining the
optimal bounds.Comment: For the proceedings of QCMC 201
Uncertainty from Heisenberg to Today
Full text linkWe explore the different meanings of "quantum uncertainty" contained in
Heisenberg's seminal paper from 1927, and also some of the precise definitions
that were explored later. We recount the controversy about "Anschaulichkeit",
visualizability of the theory, which Heisenberg claims to resolve. Moreover, we
consider Heisenberg's programme of operational analysis of concepts, in which
he sees himself as following Einstein. Heisenberg's work is marked by the
tensions between semiclassical arguments and the emerging modern quantum
theory, between intuition and rigour, and between shaky arguments and
overarching claims. Nevertheless, the main message can be taken into the new
quantum theory, and can be brought into the form of general theorems. They come
in two kinds, not distinguished by Heisenberg. These are, on one hand,
constraints on preparations, like the usual textbook uncertainty relation, and,
on the other, constraints on joint measurability, including trade-offs between
accuracy and disturbance.Comment: 36 pages, 1 figur
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