Measurement Uncertainty for Finite Quantum Observables

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Schwonnek, René; Reeb, David; Werner, Reinhard: Measurement Uncertainty for Finite Quantum Observables. In: Mathematics 4 (2016), Nr. 2, 38. DOI: https://doi.org/10.3390/math4020038

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Abstract: 
Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result x rather than y, for any pair (x,y) . This induces a notion of optimal transport cost for a pair of probability distributions, and we include an Appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a “true value” is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally-entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples.
License of this version: CC BY 4.0 Unported
Document Type: article
Publishing status: publishedVersion
Issue Date: 2016
Appears in Collections:Fakultät für Mathematik und Physik

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1 image of flag of Germany Germany 44 77.19%
2 image of flag of China China 3 5.26%
3 image of flag of Vietnam Vietnam 2 3.51%
4 image of flag of United States United States 2 3.51%
5 image of flag of United Kingdom United Kingdom 2 3.51%
6 image of flag of France France 2 3.51%
7 image of flag of Netherlands Netherlands 1 1.75%
8 image of flag of Hungary Hungary 1 1.75%

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