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An uncertainty principle for unimodular quantum groups

Jason Crann, Mehrdad Kalantar

We present a generalization of Hirschman's entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to q-traces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.We present a generalization of Hirschman's entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to q-traces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.

arXiv:1404.1276Journal of Mathematical PhysicsJul 28, 201410 citations

Uncertainty principles for locally compact quantum groups

Chunlan Jiang, Zhengwei Liu, Jinsong Wu

Journal of Functional AnalysisSep 23, 201718 citations

Entropic uncertainty relations under localizations on discrete quantum groups

Sang-Gyun Youn

The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group G is the free orthogonal quantum group ON+ with N ≥ 3 or if G admits an infinite Λ(p) set with p > 2. On the other hand, this paper explains the reason why such phenomena do not appear when G is one of the connected semisimple compact Lie groups, O2+, and the quantum SU(2) groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group G is the free orthogonal quantum group ON+ with N ≥ 3 or if G admits an infinite Λ(p) set with p > 2. On the other hand, this paper explains the reason why such phenomena do not appear when G is one of the connected semisimple compact Lie groups, O2+, and the quantum SU(2) groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.

arXiv:1804.01199Journal of Mathematical PhysicsJul 1, 20181 citation

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Steeve Zozor, Christophe Vignat

arXiv:math/0605510Physica A Statistical Mechanics and its ApplicationsOct 18, 200660 citations

Uncertainty principles associated with multi-dimensional linear canonical transform

Owais Ahmad, Azzedine Achak, Neyaz A. Sheikh +1 more

The main objective of this paper is to establish Heisenberg’s and Beckner’s uncertainty principles associated with multi-dimensional linear canonical transform.

International Journal of Geometric Methods in Modern PhysicsDec 4, 202116 citations

Uncertainty related to position and momentum localization of a quantum state

Łukasz Rudnicki

This paper presents the uncertainty related to position and momentum localization of a quantum state in terms of entropic uncertainty relations. We slightly improve the inequality given in [Phys. Rev. A 74, 052101 (2006)] and introduce a new entropic measure with corresponding uncertainty relation.

arXiv:1010.3269Oct 15, 20107 citations

Minimizers of the entropic uncertainty principle for locally compact quantum groups

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Minimizers of the entropic uncertainty principle for locally compact quantum groups

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