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arXiv:2003.02086 · Physical Review A

Entropic uncertainty relations and the quantum-to-classical transition

Isadora Veeren, Fernando de Melo

Published Aug 10, 2020• 11 citations
arXivHTMLPDF

Abstract

Our knowledge of quantum mechanics can satisfactorily describe simple microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measurements must increase in order to keep quantum properties, a desirable feature for large quantum computers.

DOI
10.1103/physreva.102.022205

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Our knowledge of quantum mechanics can satisfactorily describe simple microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measurements must increase in order to keep quantum properties, a desirable feature for large quantum computers.
Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, rather little is known about entropic uncertainty relations with more than two measurement settings. In this note we review known results and open questions. The uncertainty principle is one of the fundamental ideas of quantum mechanics. Since Heisenberg’s uncertainty relations for canonically conjugate variables, they have been one of the most prominent examples of how quantum mechanics differs from the classical world (Heisenberg, 1927). Uncertainty relations today are probably best known in the form given by (Robertson, 1929), who extended Heisenberg’s result to two arbitrary observables A and B. Robertson’s relation states that if we prepare many copies of the state |ψ〉, and measure each copy individually using either A or B, we have
Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constrains on the accuracy of two non-commuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation of coherence between two measurement non-commuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the l1l_1l1​ norm of coherence.
Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of non-local correlations. The uncertainty principle states that there exist pairs of non-compatible measurements on quantum systems such that their outcomes cannot be simultaneously predicted by any observer. Non-local correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in quantum mechanics in the presence of entanglement. Here, we show that these two essential properties of quantum mechanics are quantitatively related. Namely, we provide an entropic uncertainty relation that gives a lower bound on the uncertainty of the binary outcomes of two measurements in terms of the maximum Clauser-Horne-Shimony-Holt value that can be achieved using the same measurements. We discuss an application of this uncertainty relation to certify a quantum source using untrusted devices.
The uncertainty principle in quantum physics for non-commuting observables can be quantitatively expressed in various mathematical forms. Well known are common Heisenberg-like uncertainty relations in which the product of dispersions of two non-commuting observables is used. An alternative form of uncertainty relations represents the so-called entropic uncertainty relations in which the sum of entropies of the considered non-commuting observables is employed. The aim of this paper is to show that the entropic uncertainty relations exhibit several interesting mathematical properties and can be taken as an adequate alternative formulation of the uncertainty principle in quantum mechanics. To compare the mathematical properties of the common and entropic uncertainty relations we present some simple quantum systems with two non-commuting observables. Dedicated to Dr B Mamojka on the occasion of his fiftieth birthday. Abstrakt. Princip neurcitosti pro nekomutující veliciny v kvantové fyzice mze být matematicky vyjádren r formou. Dobre známé jsou relace neurcitosti Heisenbergova typu, v nichz se vyuzí vá soucinu disperzí dvou nekomutujících fyzikálních velicin. Alternativou jsou tzv. entropické relace neurcitosti, v nichz vystupuje soucet entropií uvazovaných velicin. Cílem tohoto clánku je ukázat, ze tyto relace mají nekolik z matematického hlediska zajímavých vlastností a mohou být pouzity k alternativnímu vyjádrení principu neurcitosti v kvantové mechanice. Srovnání vlastností Heisengergových a entropických relací neurcitosti je demonstrováno na jednoduchém kvantovém systému s dvema nekomutujícími velicinami.
How much of the uncertainty in predicting measurement outcomes for noncommuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two noncommuting observables into a classical component and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits ``purity-based'' lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a nontrivial interplay between quantum and classical randomness in any finite-dimensional state.
Abstract Reality of quantum observables, a feature of long-standing interest within foundations of quantum mechanics, has recently been quantified and deeply studied by means of entropic measures (Dieguez and Angelo 2018 Phys. Rev . A 97 022107). However, there is no state-independent ‘reality trade-off’ between non-commuting observables, as in certain systems all observables are real (Bilobran and Angelo 2015 Europhys. Lett . 112 40005). We show that the entropic uncertainty relation in the presence of quantum memory (Berta et al 2010 Nat. Phys . 6 659) perfectly supplements the discussed notion of reality, rendering trade-offs between reality and quantum uncertainty. State-independent complementarity inequalities involving entropic measures of both, uncertainty and reality, for two observables are presented.