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Entropy 2017, 19(7), 301; doi:10.3390/e19070301

Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation

1
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
2
Istituto Nazionale di Alta Matematica (INDAM-GNAMPA), 00185 Roma, Italy
3
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano, 20133 Milano, Italy
*
Author to whom correspondence should be addressed.
Received: 26 May 2017 / Revised: 21 June 2017 / Accepted: 21 June 2017 / Published: 24 June 2017
(This article belongs to the Special Issue Quantum Information and Foundations)
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Abstract

Heisenberg’s uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true’ probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements. View Full-Text
Keywords: measurement uncertainty relations; relative entropy; position; momentum measurement uncertainty relations; relative entropy; position; momentum
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Barchielli, A.; Gregoratti, M.; Toigo, A. Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation. Entropy 2017, 19, 301.

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