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Entropic measurement uncertainty relations for spin observables

Alberto Barchielli, Matteo Gregoratti

Published Dec 20, 2019• 0 citations
OpenAlex

Abstract

The information-theoretic formulation of quantum measurement uncertainty \nrelations (MURs), based on the notion of relative entropy between measurement \nprobabilities, is extended to the set of all the spin components for a generic \nspin $s$. For a physical class of approximate joint measurements of the spin \ncomponents, we define the device information loss as the maximum loss of \ninformation per observable occurring in approximating the ideal incompatible \ncomponents with the joint measurement at hand. By optimizing on the measuring \ndevice, we define the notion of minimum information loss. By using this \nnotions, we show how to give a significant formulation of state independent \nMURs in the case of infinitely many target observables. The same construction \nworks as well for finitely many observables, and we study the related MURs for \ntwo and three orthogonal spin components. The minimum information loss plays \nalso the role of measure of incompatibility and in this respect it allows us to \ncompare quantitatively the incompatibility of various sets of spin observables, \nwith different number of involved components and different values of $s$.

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The information-theoretic formulation of quantum measurement uncertainty \nrelations (MURs), based on the notion of relative entropy between measurement \nprobabilities, is extended to the set of all the spin components for a generic \nspin sss. For a physical class of approximate joint measurements of the spin \ncomponents, we define the device information loss as the maximum loss of \ninformation per observable occurring in approximating the ideal incompatible \ncomponents with the joint measurement at hand. By optimizing on the measuring \ndevice, we define the notion of minimum information loss. By using this \nnotions, we show how to give a significant formulation of state independent \nMURs in the case of infinitely many target observables. The same construction \nworks as well for finitely many observables, and we study the related MURs for \ntwo and three orthogonal spin components. The minimum information loss plays \nalso the role of measure of incompatibility and in this respect it allows us to \ncompare quantitatively the incompatibility of various sets of spin observables, \nwith different number of involved components and different values of sss.
Abstract The information-theoretic formulation of quantum measurement uncertainty relations (MURs), based on the notion of relative entropy between measurement probabilities, is extended to the set of all the spin components for a generic spin s. For an approximate measurement of a spin vector, which gives approximate joint measurements of the spin components, we define the device information loss as the maximum loss of information per observable occurring in approximating the ideal incompatible components with the joint measurement at hand. By optimizing on the measuring device, we define the notion of minimum information loss. By using these notions, we show how to give a significant formulation of state independent MURs in the case of infinitely many target observables. The same construction works as well for finitely many observables, and we study the related MURs for two and three orthogonal spin components. The minimum information loss plays also the role of measure of incompatibility and in this respect it allows us to compare quantitatively the incompatibility of various sets of spin observables, with different number of involved components and different values of s.
We formulate entropic measurements uncertainty relations (MURs) for a spin-1/2 system. When incompatible observables are approximatively jointly measured, we use relative entropy to quantify the information lost in approximation and we prove positive lower bounds for such a loss: there is an unavoidable information loss. Firstly we allow only for covariant approximate joint measurements and we find state-dependent MURs for two or three orthogonal spin-1/2 components. Secondly we consider any possible approximate joint measurement and we find state-independent MURs for two or three spin-1/2 components. In particular we study how MURs depend on the angle between two spin directions. Finally, we extend our approach to infinitely many incompatible observables, namely to the spin components in all possible directions. In every scenario, we always consider also the characterization of the optimal approximate joint measurements.
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.
We establish rigorously in this work a measurement uncertainty relation (MUR) for three unbiased qubit observables, which was previously shown to hold true under some presumptions. The triplet MUR states that the uncertainty, which is quantified by the total statistical distance between the target observables and the jointly implemented observables, is lower bounded by an incompatibility measure that reflects the joint measurement conditions. We derive a necessary and sufficient condition for the triplet MUR to be saturated and the corresponding optimal measurement. To facilitate experimental tests of MURs we propose a straightforward implementation of the optimal joint measurements. Lastly, by a symmetry argument, the exact values of incompatibility measure are analytically calculated for some symmetric triplets. We anticipate that our work may enrich the understanding of quantum incompatibility in terms of MURs and inspire further applications in quantum information science. This work presents a complete theory relevant to a parallel work [Y.-L. Mao et al., Testing Heisenberg-type measurement uncertainty relations of three observables, Phys. Rev. Lett. 131, 150203 (2023)] on experimental tests.
We investigate entropic uncertainty relations for two or more binary measurements, for example, spin-12\frac{1}{2}21​ or polarization measurements. We argue that the effective anticommutators of these measurements, i.e., the anticommutators evaluated on the state prior to measuring, are an expedient measure of measurement incompatibility. Based on the knowledge of pairwise effective anticommutators we derive a class of entropic uncertainty relations in terms of conditional R\'enyi entropies. Our uncertainty relations are formulated in terms of effective measures of incompatibility, which can be certified in a device-independent fashion. Consequently, we discuss potential applications of our findings to device-independent quantum cryptography. Moreover, to investigate the tightness of our analysis we consider the simplest (and very well studied) scenario of two measurements on a qubit. We find that our results outperform the celebrated bound due to Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] and provide an analytical expression for the minimum uncertainty which also outperforms some recent bounds based on majorization.
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.