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Applied Sciences

Measurements of Entropic Uncertainty Relations in Neutron Optics

Bülent Demirel, Stephan Sponar, Yuji Hasegawa

Published Feb 6, 2020• 6 citations
Publisher

Abstract

The emergence of the uncertainty principle has celebrated its 90th anniversary recently. For this occasion, the latest experimental results of uncertainty relations quantified in terms of Shannon entropies are presented, concentrating only on outcomes in neutron optics. The focus is on the type of measurement uncertainties that describe the inability to obtain the respective individual results from joint measurement statistics. For this purpose, the neutron spin of two non-commuting directions is analyzed. Two sub-categories of measurement uncertainty relations are considered: noise–noise and noise–disturbance uncertainty relations. In the first case, it will be shown that the lowest boundary can be obtained and the uncertainty relations be saturated by implementing a simple positive operator-valued measure (POVM). For the second category, an analysis for projective measurements is made and error correction procedures are presented.

DOI
10.3390/app10031087

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Comment on "Uncertainty Relations for Positive Operator Valued Measures"

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The principal aim of this Comment is to correct those entropic uncertainty relations that are presented in a paper by Massar [arXiv:quant-ph/0703036v2 (current version)], concerning two approaches to a study of the noise produced by POVM's. It is next emphasized that the first of the entropic bounds for POVM obtained by the above author has been already presented in Ref. [8]. Some exposition obscurity with equation (14) of the commented paper is elucidated. Finally, some more specific remarks on the paper content are given.

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arXiv:quant-ph/0703036•Physical Review A•Oct 17, 2007•31 citations
The emergence of the uncertainty principle has celebrated its 90th anniversary recently. For this occasion, the latest experimental results of uncertainty relations quantified in terms of Shannon entropies are presented, concentrating only on outcomes in neutron optics. The focus is on the type of measurement uncertainties that describe the inability to obtain the respective individual results from joint measurement statistics. For this purpose, the neutron spin of two non-commuting directions is analyzed. Two sub-categories of measurement uncertainty relations are considered: noise–noise and noise–disturbance uncertainty relations. In the first case, it will be shown that the lowest boundary can be obtained and the uncertainty relations be saturated by implementing a simple positive operator-valued measure (POVM). For the second category, an analysis for projective measurements is made and error correction procedures are presented.
The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theoretical analysis predicts that general quantum measurements are necessary to saturate some such uncertainty relations and thereby overcome certain limitations of projective measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin- 1 / 2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e. positive-operator valued measures) as predicted theoretically. These results showcase experimentally the advantage obtainable by general quantum measurements over projective measurements when probing certain uncertainty relations.
Information-theoretic uncertainty relations formulate the joint immeasurability of two noncommuting observables in terms of information entropies.The tradeoff of the accuracy in the outcome of two successive measurements manifests in entropic noise-disturbance uncertainty relations.Recent theoretical analysis predicts that projective measurements are not optimal, with respect to the noise-disturbance tradeoffs.Therefore, the results in our previous paper [Phys. Rev. Lett. 115, 030401 (2015)] are outperformed by general quantum measurements.Here, we experimentally test a tight information-theoretic measurement uncertainty relation for three-outcome positive-operator-valued measures, using neutron spin-1 2 qubits.The obtained results violate the lower bound for projective measurements as theoretically predicted.
Information-theoretic definitions for noise and disturbance in quantum measurements were given in [Phys. Rev. Lett. 112, 050401 (2014)] and a state-independent noise-disturbance uncertainty relation was obtained. Here, we derive a tight noise-disturbance uncertainty relation for complementary qubit observables and carry out an experimental test. Successive projective measurements on the neutron's spin-1/2 system, together with a correction procedure which reduces the disturbance, are performed. Our experimental results saturate the tight noise-disturbance uncertainty relation for qubits when an optimal correction procedure is applied.
Heisenberg’s uncertainty principle in a formulation of uncertainties, intrinsic to any quantum system, is rigorously proven and demonstrated in various quantum systems. Nevertheless, Heisenberg’s original formulation of the uncertainty principle was given in terms of a reciprocal relation between the error of a position measurement and the thereby induced disturbance on a subsequent momentum measurement. However, a naive generalization of a Heisenberg-type error-disturbance relation for arbitrary observables is not valid. An alternative universally valid relation was derived by Ozawa in 2003. Though universally valid, Ozawa’s relation is not optimal. Recently, Branciard has derived a tight error-disturbance uncertainty relation (EDUR), describing the optimal trade-off between error and disturbance under certain conditions. Here, we report a neutron-optical experiment that records the error of a spin-component measurement, as well as the disturbance caused on another spin-component to test EDURs. We demonstrate that Heisenberg’s original EDUR is violated, and Ozawa’s and Branciard’s EDURs are valid in a wide range of experimental parameters, as well as the tightness of Branciard’s relation.
The principal aim of this Comment is to correct those entropic uncertainty relations that are presented in a paper by Massar [arXiv:quant-ph/0703036v2 (current version)], concerning two approaches to a study of the noise produced by POVM's. It is next emphasized that the first of the entropic bounds for POVM obtained by the above author has been already presented in Ref. [8]. Some exposition obscurity with equation (14) of the commented paper is elucidated. Finally, some more specific remarks on the paper content are given.
How much unavoidable randomness is generated by a positive-operator-valued measure (POVM)? We address this question using two complementary approaches. First, we study the variance of a real variable associated with the POVM outcomes. In this context we introduce an uncertainty operator which measures how much additional noise is introduced by carrying out a POVM rather than a von Neumann measurement. We illustrate this first approach by studying the variances of joint estimates of ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{z}$ for spin-1∕21∕21∕2 particles. We show that for unbiased measurements the sum of these variances is lower bounded by 111. In our second approach we study the entropy of the POVM outcomes. In particular, we try to establish lower bounds on the entropy of the POVM outcomes. We illustrate this second approach by examples.