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Journal of Physics A Mathematical and Theoretical

Uncertainty relations for multiple operators without covariances

Bin Chen, Pan Lian

Published Jan 19, 2022• 5 citations
Publisher

Abstract

Abstract In this paper, we prove the sum and product uncertainty relations conjectured by V Dodonov for multiple observables. The uncertainty relations for linear combinations of position and momentum recently obtained by Kechrimparis and Weigert are recovered. Furthermore, the entropic uncertainty relations conjectured by the latter authors are proved for specific cases. At last, we revisit the uncertainty relation for triple canonical operators and obtain a tighter bound on real Hilbert space. A quantitative stability result is given as well.

DOI
10.1088/1751-8121/ac4ce8

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Abstract In this paper, we prove the sum and product uncertainty relations conjectured by V Dodonov for multiple observables. The uncertainty relations for linear combinations of position and momentum recently obtained by Kechrimparis and Weigert are recovered. Furthermore, the entropic uncertainty relations conjectured by the latter authors are proved for specific cases. At last, we revisit the uncertainty relation for triple canonical operators and obtain a tighter bound on real Hilbert space. A quantitative stability result is given as well.
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Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.
Abstract Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position x and momentum p . This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take x - p correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to x and p , but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that takes correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.