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:1803.08720

Unified Variance-based Uncertainty Relation.

Xiao Zheng, Shao‐Qiang Ma, Guo‐Feng Zhang, Heng Fan, Wu‐Ming Liu

Published Mar 23, 2018• 0 citations
arXivHTMLPDF

Abstract

We unify the product and the sum form variance-based uncertainties to a Schr\{o}dinger-liked uncertainty relation. Some well-known uncertainties in either product or sum form can be recovered by the Schr\{o}dinger-liked uncertainty, i.e., the product form uncertainty is the Schr\{o}dinger-liked uncertainty relation for Hermitian operators and the sum form uncertainty is actually Schr\{o}dinger-liked uncertainty relation for non-Hermitian operators. Thus the Schr\{o}dinger-liked uncertainty relation provides a unified uncertainty relation. Also, the unified uncertainty reveals some quantum properties that the traditional uncertainty relation cannot do, and indicates that the uncertainty relation for the incompatible observables is bounded not only by the commutation relation between themselves, but also by the quantities related with the other operator, which is named as auxiliary operator. The importance of the unified uncertainty, as well as the concept of the auxiliary operator deduced from it, is demonstrated in explaining and fixing the trivial lower bound appearing in the traditional uncertainty relations.

Related papers

Unified and Exact Framework for Variance-Based Uncertainty Relations

Xiao Hong Zheng, Shao‐Qiang Ma, Guo‐Feng Zhang +2 more

We provide a unified and exact framework for the variance-based uncertainty relations. This unified framework not only recovers some well-known previous uncertainty relations, but also fixes the deficiencies of them. Utilizing the unified framework, we can construct the new uncertainty relations in both product and sum form for two and more incompatible observables with any tightness we require. Moreover, one can even construct uncertainty equalities to exactly express the uncertainty relation by the unified framework, and the framework is therefore exact in describing the uncertainty relation. Some applications have been provided to illustrate the importance of this unified and exact framework. Also, we show that the contradiction between uncertainty relation and non-Hermitian operator, i.e., most of uncertainty relations will be violated when applied to non-Hermitian operators, can be fixed by this unified and exact framework.

Scientific Reports•Jan 10, 2020•17 citations

Stronger sum uncertainty relations for non-Hermitian operators

Xiao-Feng Song, Yi-Fang Ren, Shuang Liu +2 more

The uncertainty relations (URs) of two arbitrary Hermitian and non-Hermitian incompatible operators represented by the product of variances have been confirmed theoretically and experimentally in various physical systems. However, the lower bound of the product uncertainty inequality can be null even for two noncommuting operators, i.e., a trivial case. Therefore, for two incompatible operators over the measured system state, the associated URs regarding the sum of variances are valid in a state-dependent manner, and the lower bound is guaranteed to be nontrivial. Although the sum URs formulated for Hermitian and unitary operators have been affirmed, the general forms for arbitrary non-Hermitian operators have not yet been investigated. This study presents the sum URs for non-Hermitian operators acting on system states using an appropriate Hilbert-space metric. The compatible forms of our sum inequalities with the conventional quantum mechanics are also provided via the $G$-metric formalism. Concrete examples illustrate the validity of the proposed sum URs in both $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phases. The developed methods and results can help give an in-depth understanding of the usefulness of $G$-metric formalism in non-Hermitian quantum mechanics and the sum URs of incompatible operators within.

arXiv:2407.20481•Physical Review A•Jan 29, 2025•2 citations

Strong Variance-Based Uncertainty Relations and Uncertainty Intervals

Yunlong Xiao, Naihuan Jing, Bing Yu +2 more

Uncertainty relations occupy a fundamental position in quantum mechanics. We propose stronger variance-based uncertainty relations for the product and sum of variances of two incompatible observables in a finite dimensional Hilbert space. It is shown that the new uncertainty relations provide near-optimal state-dependent bounds, which can be useful for quantum metrology, entanglement detection etc. in quantum information theory. It is further shown that the uncertainty relations are related to the spreads of the distribution of measurement outcomes caused by incompatible observables. Intuitively, this means that the ability of learning the distribution has both the upper and lower bounds. Combination of these bounds provides naturally an uncertainty interval which captures the essence of uncertainty in quantum theory. Finally, we explain how to employ entropic uncertainty relations to derive lower bounds for the product of variances of incompatible observables.

•Oct 5, 2016•0 citations

A Stronger Multi-observable Uncertainty Relation

Qiu-Cheng Song, Jun-Li Li, Guang-Xiong Peng +1 more

Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization of an existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones for spin- and spin-1 particles indicate that the obtained uncertainty relation gives a better lower bound.

arXiv:1701.01072•Scientific Reports•Mar 20, 2017•47 citations

Tighter uncertainty and reverse uncertainty relations

Debasis Mondal, Shrobona Bagchi, Arun Kumar Pati

We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.

arXiv:1607.06712•Physical Review A•May 18, 2017•54 citations

Near-Optimal Variance-Based Uncertainty Relations

Yunlong Xiao, Naihuan Jing, Bing Yu +2 more

Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.

arXiv:1610.01692•Frontiers in Physics•Apr 1, 2022•8 citations
We unify the product and the sum form variance-based uncertainties to a Schr{o}dinger-liked uncertainty relation. Some well-known uncertainties in either product or sum form can be recovered by the Schr{o}dinger-liked uncertainty, i.e., the product form uncertainty is the Schr{o}dinger-liked uncertainty relation for Hermitian operators and the sum form uncertainty is actually Schr{o}dinger-liked uncertainty relation for non-Hermitian operators. Thus the Schr{o}dinger-liked uncertainty relation provides a unified uncertainty relation. Also, the unified uncertainty reveals some quantum properties that the traditional uncertainty relation cannot do, and indicates that the uncertainty relation for the incompatible observables is bounded not only by the commutation relation between themselves, but also by the quantities related with the other operator, which is named as auxiliary operator. The importance of the unified uncertainty, as well as the concept of the auxiliary operator deduced from it, is demonstrated in explaining and fixing the trivial lower bound appearing in the traditional uncertainty relations.
We provide a unified and exact framework for the variance-based uncertainty relations. This unified framework not only recovers some well-known previous uncertainty relations, but also fixes the deficiencies of them. Utilizing the unified framework, we can construct the new uncertainty relations in both product and sum form for two and more incompatible observables with any tightness we require. Moreover, one can even construct uncertainty equalities to exactly express the uncertainty relation by the unified framework, and the framework is therefore exact in describing the uncertainty relation. Some applications have been provided to illustrate the importance of this unified and exact framework. Also, we show that the contradiction between uncertainty relation and non-Hermitian operator, i.e., most of uncertainty relations will be violated when applied to non-Hermitian operators, can be fixed by this unified and exact framework.
The uncertainty relations (URs) of two arbitrary Hermitian and non-Hermitian incompatible operators represented by the product of variances have been confirmed theoretically and experimentally in various physical systems. However, the lower bound of the product uncertainty inequality can be null even for two noncommuting operators, i.e., a trivial case. Therefore, for two incompatible operators over the measured system state, the associated URs regarding the sum of variances are valid in a state-dependent manner, and the lower bound is guaranteed to be nontrivial. Although the sum URs formulated for Hermitian and unitary operators have been affirmed, the general forms for arbitrary non-Hermitian operators have not yet been investigated. This study presents the sum URs for non-Hermitian operators acting on system states using an appropriate Hilbert-space metric. The compatible forms of our sum inequalities with the conventional quantum mechanics are also provided via the GGG-metric formalism. Concrete examples illustrate the validity of the proposed sum URs in both PT\mathcal{PT}PT-symmetric and PT\mathcal{PT}PT-broken phases. The developed methods and results can help give an in-depth understanding of the usefulness of GGG-metric formalism in non-Hermitian quantum mechanics and the sum URs of incompatible operators within.
Uncertainty relations occupy a fundamental position in quantum mechanics. We propose stronger variance-based uncertainty relations for the product and sum of variances of two incompatible observables in a finite dimensional Hilbert space. It is shown that the new uncertainty relations provide near-optimal state-dependent bounds, which can be useful for quantum metrology, entanglement detection etc. in quantum information theory. It is further shown that the uncertainty relations are related to the spreads of the distribution of measurement outcomes caused by incompatible observables. Intuitively, this means that the ability of learning the distribution has both the upper and lower bounds. Combination of these bounds provides naturally an uncertainty interval which captures the essence of uncertainty in quantum theory. Finally, we explain how to employ entropic uncertainty relations to derive lower bounds for the product of variances of incompatible observables.
Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization of an existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones for spin- and spin-1 particles indicate that the obtained uncertainty relation gives a better lower bound.
We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.