Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1404.1276 · Journal of Mathematical Physics

An uncertainty principle for unimodular quantum groups

Jason Crann, Mehrdad Kalantar

Published Jul 28, 2014• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4890288

Related papers

Minimizers of the entropic uncertainty principle for locally compact quantum groups

Jinsong Wu

In this paper, we give an alternative proof of the Hirschman-Beckner uncertainty principle for locally compact quantum groups and characterize the minimizers of the uncertainty principles.

Journal of Mathematical Physics•Aug 1, 2019•0 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.

arXiv:1804.01199•Journal of Mathematical Physics•Jul 1, 2018•1 citation

Continuous Krishna–Parthasarathy entropic uncertainty principle

K. Mahesh Krishna

In 2002, Krishna and Parthasarathy7 derived discrete quantum version of Maassen–Uffink8 entropic uncertainty principle. In this paper, using the notion of continuous operator-valued frames, we derive an entropic uncertainty principle for arbitrary family of operators indexed by measure spaces having finite measure. We give an application to the special case of compact groups.

arXiv:2405.08003•Infinite Dimensional Analysis Quantum Probability and Related Topics•Apr 19, 2024•0 citations

Asymptotic entropic uncertainty relations

Radosław Adamczak, Rafał Latała, Zbigniew Puchała +1 more

We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.

arXiv:1412.7065•Journal of Mathematical Physics•Mar 1, 2016•18 citations

Quantum complementarity through entropic certainty principles

Javier M. Magán, Diego Pontello

We approach the physical implications of the noncommutative nature of complementary observable algebras (COAs) from an information theoretic perspective. In particular, we derive a general entropic certainty principle stating that the sum of two relative entropies, naturally related to the COA, is equal to the so-called algebraic index of the associated inclusion. Uncertainty relations then arise by monotonicity of the relative entropies that participate in the underlying entropic certainty. Examples and applications are described in quantum field theories with global symmetries, where the COAs are formed by the charge-anticharge local operators (intertwiners) and the unitary representations of the symmetry group (twists), and in theories with local symmetries, where the COAs are formed by Wilson and t'Hooft loops. In general, the entropic certainty principle naturally captures the physics of order and disorder parameters, a feature that makes it a generic handle for the information theoretic characterization of quantum phases.

arXiv:2005.01760•Physical Review A•Jan 15, 2021•27 citations

Entropic distinguishability of quantum fields in phase space

Sara Ditsch, Tobias Haas

We present a general way of quantifying the entropic uncertainty of quantum field configurations in phase space in terms of entropic distinguishability with respect to the vacuum. Our approach is based on the functional Husimi Q -distribution and a suitably chosen relative entropy, which we show to be non-trivially bounded from above by the uncertainty principle. The resulting relative entropic uncertainty relation is as general as the concept of coherent states and thus holds for quantum fields of bosonic and fermionic type. Its simple form enables diverse applications, among which we present a complete characterization of the uncertainty surplus of arbitrary states in terms of the total particle number for a scalar field and the fermionic description of the Ising model. Moreover, we provide a quantitative interpretation of the role of the uncertainty principle for quantum phase transitions.

arXiv:2307.06128•Quantum•Jul 17, 2024•6 citations
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.
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.
We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.
We approach the physical implications of the noncommutative nature of complementary observable algebras (COAs) from an information theoretic perspective. In particular, we derive a general entropic certainty principle stating that the sum of two relative entropies, naturally related to the COA, is equal to the so-called algebraic index of the associated inclusion. Uncertainty relations then arise by monotonicity of the relative entropies that participate in the underlying entropic certainty. Examples and applications are described in quantum field theories with global symmetries, where the COAs are formed by the charge-anticharge local operators (intertwiners) and the unitary representations of the symmetry group (twists), and in theories with local symmetries, where the COAs are formed by Wilson and t'Hooft loops. In general, the entropic certainty principle naturally captures the physics of order and disorder parameters, a feature that makes it a generic handle for the information theoretic characterization of quantum phases.
We present a general way of quantifying the entropic uncertainty of quantum field configurations in phase space in terms of entropic distinguishability with respect to the vacuum. Our approach is based on the functional Husimi Q -distribution and a suitably chosen relative entropy, which we show to be non-trivially bounded from above by the uncertainty principle. The resulting relative entropic uncertainty relation is as general as the concept of coherent states and thus holds for quantum fields of bosonic and fermionic type. Its simple form enables diverse applications, among which we present a complete characterization of the uncertainty surplus of arbitrary states in terms of the total particle number for a scalar field and the fermionic description of the Ising model. Moreover, we provide a quantitative interpretation of the role of the uncertainty principle for quantum phase transitions.