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:1504.05725 · Physical Review A

Role of quantum non-Gaussian distance in entropic uncertainty relations

Wonmin Son

Published Jul 17, 2015• 14 citations
arXivHTMLPDF

Abstract

A Gaussian distribution of a quantum state with continuous spectra is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{p}$ are featured explicitly. From the construction of the entropic uncertainty relation, it follows in a transparent manner that (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modified form, ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}\ensuremath{\hbar}{e}^{\mathcal{N}}/2$; (ii) the additional factor $\mathcal{N}$ quantifies the quantum non-Gaussianity of the probability distributions of two observables; and (iii) the lower bound of the entropic uncertainty relation for a non-Gaussian continuous-variable (CV) mixed state becomes stronger with purity. The optimality of specific non-Gaussian CV states for the refined uncertainty relation has been investigated and the existence of a new class of CV quantum state is identified.

DOI
10.1103/physreva.92.012114

Related papers

Continuous-variable entropic uncertainty relations

Anaelle Hertz, Nicolas J. Cerf

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.

arXiv:1809.01052•Journal of Physics A Mathematical and Theoretical•Feb 1, 2019•76 citations

Entropy-power uncertainty relations: towards a tight inequality for all Gaussian pure states

Anaelle Hertz, Michael G Jabbour, Nicolas J. Cerf

Abstract We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity. Interestingly, it implies the generalized (rotation-invariant) Schrödinger–Robertson uncertainty relation exactly as the original entropy-power uncertainty relation implies Heisenberg relation. It is saturated for all Gaussian pure states, in contrast with hitherto known entropic formulations of the uncertainty principle.

arXiv:1702.07286•Journal of Physics A Mathematical and Theoretical•Aug 9, 2017•18 citations

Non-Gaussianity and entropy-bounded uncertainty relations: Application to detection of non-Gaussian entangled states

Kyunghyun Baek, Hyunchul Nha

We suggest an improved version of the Robertson--Schr\"odinger uncertainty relation for canonically conjugate variables by taking into account a pair of characteristics of states: non-Gaussianity and mixedness quantified by using fidelity and entropy, respectively. This relation is saturated by both Gaussian and Fock states and provides a strictly improved bound for any non-Gaussian states or mixed states. For the case of Gaussian states, it is reduced to the entropy-bounded uncertainty relation derived by Dodonov. Furthermore, we consider readily computable measures of both characteristics and find a weaker but more readily accessible bound. With its generalization to the case of two-mode states, we show applicability of the relation to detect entanglement of non-Gaussian states.

arXiv:1811.10207•Physical Review A•Oct 10, 2018•14 citations

Entropic uncertainty relations – A survey

Stephanie Wehner, Andreas Winter

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

•Jan 1, 2009•239 citations

Uncertainty, entropy, and non-Gaussianity for mixed states

Aikaterini Mandilara, Evgueni Karpov, Nicolas J. Cerf

In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated. Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded uncertainty relation by adding an additional parameter characterizing the state, namely its degree of non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation. The common points and differences between the two extensions of the uncertainty relation help us to draw more general conclusions concerning the bounds on the non-Gaussianity of mixed states.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 30, 2010•2 citations

Tight Entropic Uncertainty Relations

Alberto Riccardi, Lorenzo Maccone

Entropic uncertainty relations $H(A)+H(B)\geqslant γ$ give a nonzero lower bound $γ$ to the sum of the Shannon entropies $H$ of the outcome probabilities of incompatible observables $A$ and $B$. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds $γ$ typically are state independent. Here we provide a state-independent lower bound $γ_s$ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter $s\to 2$, becomes asymptotically tight for all $A,B$. The bound can be extended to Renyi entropies. After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/

quant-pharXiv:2605.01832•May 5, 2026•0 citations
A Gaussian distribution of a quantum state with continuous spectra is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables x\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}x\ifmmode^\elseˆ\fi and p\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}p\ifmmode^\elseˆ\fi​, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations \ensuremathσx{\ensuremath{\sigma}}_{x}\ensuremathσx​ and \ensuremathσp{\ensuremath{\sigma}}_{p}\ensuremathσp​ are featured explicitly. From the construction of the entropic uncertainty relation, it follows in a transparent manner that (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modified form, \ensuremathσx\ensuremathσp\ensuremath≥\ensuremathℏeN/2{\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}\ensuremath{\hbar}{e}^{\mathcal{N}}/2\ensuremathσx​\ensuremathσp​\ensuremath≥\ensuremathℏeN/2; (ii) the additional factor N\mathcal{N}N quantifies the quantum non-Gaussianity of the probability distributions of two observables; and (iii) the lower bound of the entropic uncertainty relation for a non-Gaussian continuous-variable (CV) mixed state becomes stronger with purity. The optimality of specific non-Gaussian CV states for the refined uncertainty relation has been investigated and the existence of a new class of CV quantum state is identified.
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.
Abstract We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity. Interestingly, it implies the generalized (rotation-invariant) Schrödinger–Robertson uncertainty relation exactly as the original entropy-power uncertainty relation implies Heisenberg relation. It is saturated for all Gaussian pure states, in contrast with hitherto known entropic formulations of the uncertainty principle.
We suggest an improved version of the Robertson--Schr\"odinger uncertainty relation for canonically conjugate variables by taking into account a pair of characteristics of states: non-Gaussianity and mixedness quantified by using fidelity and entropy, respectively. This relation is saturated by both Gaussian and Fock states and provides a strictly improved bound for any non-Gaussian states or mixed states. For the case of Gaussian states, it is reduced to the entropy-bounded uncertainty relation derived by Dodonov. Furthermore, we consider readily computable measures of both characteristics and find a weaker but more readily accessible bound. With its generalization to the case of two-mode states, we show applicability of the relation to detect entanglement of non-Gaussian states.
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
In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated. Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded uncertainty relation by adding an additional parameter characterizing the state, namely its degree of non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation. The common points and differences between the two extensions of the uncertainty relation help us to draw more general conclusions concerning the bounds on the non-Gaussianity of mixed states.
Entropic uncertainty relations H(A)+H(B)⩾γH(A)+H(B)\geqslant γH(A)+H(B)⩾γ give a nonzero lower bound γγγ to the sum of the Shannon entropies HHH of the outcome probabilities of incompatible observables AAA and BBB. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds γγγ typically are state independent. Here we provide a state-independent lower bound γsγ_sγs​ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter s→2s\to 2s→2, becomes asymptotically tight for all A,BA,BA,B. The bound can be extended to Renyi entropies. After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/