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

Journal of Physics A Mathematical and Theoretical

Rényi formulation of the entropic uncertainty principle for POVMs

Alexey E. Rastegin

Published Mar 25, 2010• 35 citations
Publisher

Abstract

Uncertainty relations for a pair of arbitrary measurements are posed in the form of inequalities using the Rényi entropies. The formulation deals with POVM measurements on finite-dimensional Hilbert space. Both the entropic uncertainty relations with state-dependent and state-independent bounds are presented. The results are illustrated on the example of measurements for state discrimination.

DOI
10.1088/1751-8113/43/15/155302

Related papers

Uncertainty relations for arbitrary measurement in terms of Renyi entropies

Alexey E. Rastegin

Uncertainty relations for a pair of arbitrary measurements and for a single measurement are posed in the form of inequalities using the Renyi entropies. The formulation deals with discrete observables. Both the relations with state-dependent and state-independent bounds are presented. The entropic bounds are illustrated within the distinction between non-orthogonal states.

arXiv:0805.1777•May 13, 2008•6 citations

Statement of uncertainty principle for quantum measurements in terms of the Renyi entropies

Alexey E. Rastegin

The aim of the work is to give the explicit proofs of the Renyi-entropy uncertainty relations presented in the previous work [A. Rastegin, arXiv:0805.1777]. The relations with both the state-dependent and state-independent entropic bounds are proved. For a pair of POVM measurements the two relations are obtained. The first of them is generalization of the known results, whereas the second is quite alternative. It is shown that both these relations are meaty. The important case of POVM's with one-rank elements is extra discussed. The measurements designed for distinction between two non-orthogonal quantum states are considered as examples.

arXiv:0807.2691•Jul 17, 2008•3 citations

General entropy-like uncertainty relations in finite dimensions

Steeve Zozor, G M Bosyk, M Portesi

We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B , acting on finite dimensional Hilbert space. Salicrú generalized -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet with c A (respectively c B ) being the overlap between the elements of the POVM A (respectively B ) and the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110 ) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.

arXiv:1311.5602•Journal of Physics A Mathematical and Theoretical•Nov 20, 2014•64 citations

On a generalized entropic uncertainty relation in the case of the qubit

Steeve Zozor, G. M. Bosyk, M. Portesi

We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α, β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz–Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square

•Jan 1, 2013•14 citations

Rényi formulation of uncertainty relations for POVMs assigned to a quantum design

Alexey E. Rastegin

Abstract Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and Rényi entropies for POVMs assigned to a quantum design. Relations of the Landau–Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.

arXiv:2004.05576•Journal of Physics A Mathematical and Theoretical•Jul 23, 2020•12 citations

On a generalized entropic uncertainty relation in the case of the qubit

Steeve Zozor, G M Bosyk, M Portesi

We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as an uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α, β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz–Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square [0 , 1/2] 2 in the α–β plane, and a semi-analytical expression on the line β = α. It is seen that previous results are included as particular cases. Moreover we present a semi-analytical, suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.

arXiv:1306.0409•Journal of Physics A Mathematical and Theoretical•Nov 1, 2013•23 citations
Uncertainty relations for a pair of arbitrary measurements are posed in the form of inequalities using the Rényi entropies. The formulation deals with POVM measurements on finite-dimensional Hilbert space. Both the entropic uncertainty relations with state-dependent and state-independent bounds are presented. The results are illustrated on the example of measurements for state discrimination.
The aim of the work is to give the explicit proofs of the Renyi-entropy uncertainty relations presented in the previous work [A. Rastegin, arXiv:0805.1777]. The relations with both the state-dependent and state-independent entropic bounds are proved. For a pair of POVM measurements the two relations are obtained. The first of them is generalization of the known results, whereas the second is quite alternative. It is shown that both these relations are meaty. The important case of POVM's with one-rank elements is extra discussed. The measurements designed for distinction between two non-orthogonal quantum states are considered as examples.
We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B , acting on finite dimensional Hilbert space. Salicrú generalized -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet with c A (respectively c B ) being the overlap between the elements of the POVM A (respectively B ) and the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110 ) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.
We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α, β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz–Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square
Abstract Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and Rényi entropies for POVMs assigned to a quantum design. Relations of the Landau–Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.
We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as an uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α, β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz–Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square [0 , 1/2] 2 in the α–β plane, and a semi-analytical expression on the line β = α. It is seen that previous results are included as particular cases. Moreover we present a semi-analytical, suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.