Heisenberg uncertainty for qubit measurements

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Busch, P.; Lahti, P.; Werner, R.F.: Heisenberg uncertainty for qubit measurements. In: Physical Review A 89 (2014), Nr. 1, 012129. DOI: https://doi.org/10.1103/PhysRevA.89.012129

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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 Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of ±1-valued observables A,B are tightly bounded from below by a quantity that measures the degree of incompatibility of A and B. The claim of a violation of Heisenberg is shown to fail because it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.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 Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of ±1-valued observables A,B are tightly bounded from below by a quantity that measures the degree of incompatibility of A and B. The claim of a violation of Heisenberg is shown to fail because it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.
License of this version: Es gilt deutsches Urheberrecht. Das Dokument darf zum eigenen Gebrauch kostenfrei genutzt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.
Document Type: Article
Publishing status: publishedVersion
Issue Date: 2014
Appears in Collections:Fakultät für Mathematik und Physik

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1 image of flag of United States United States 23 27.38%
2 image of flag of China China 12 14.29%
3 image of flag of Germany Germany 11 13.10%
4 image of flag of No geo information available No geo information available 9 10.71%
5 image of flag of Korea, Republic of Korea, Republic of 6 7.14%
6 image of flag of United Kingdom United Kingdom 4 4.76%
7 image of flag of France France 4 4.76%
8 image of flag of Italy Italy 3 3.57%
9 image of flag of Ireland Ireland 3 3.57%
10 image of flag of Hong Kong Hong Kong 3 3.57%
    other countries 6 7.14%

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