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General and Stronger Uncertainty Relation

Vishnu M. Bannur

Recently, Maccone and Pati [Phys. Rev. Lett. {\bf 113}, 260401 (2014)] derived few inequalities among variances of incompatible operators which they called stronger uncertainty relations, stronger than Heisenberg-Robertson or Schrodinger uncertainty relations. Here we generalize their study to get infinite number of such inequalities and propose that only one of them may be the correct uncertainty relation between incompatible operators. We get back well known uncertainty relations of Heisenberg-Robertson and Schrodinger under certain limits. We also reexamine the conclusions of Maccone and Pati and comment on their work.Recently, Maccone and Pati [Phys. Rev. Lett. {\bf 113}, 260401 (2014)] derived few inequalities among variances of incompatible operators which they called stronger uncertainty relations, stronger than Heisenberg-Robertson or Schrodinger uncertainty relations. Here we generalize their study to get infinite number of such inequalities and propose that only one of them may be the correct uncertainty relation between incompatible operators. We get back well known uncertainty relations of Heisenberg-Robertson and Schrodinger under certain limits. We also reexamine the conclusions of Maccone and Pati and comment on their work.

arXiv:1503.00405Mar 2, 20152 citations

Unified Derivation of Uncertainty Relations and Their Saturation Conditions

Chi-Kwong Li, Mikio Nakahara

We analyze uncertainty relations due to Kennard, Robertson, Schrödinger, Maccone and Pati in a unified way from matrix theory point of view. Short proofs are given to these uncertainty relations and characterizations of the saturation conditions are given.

arXiv:2302.06002Feb 12, 20230 citations

Stronger Uncertainty Relations for All Incompatible Observables

Lorenzo Maccone, Arun Kumar Pati

The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.

arXiv:1407.0338Physical Review LettersDec 31, 2014288 citations

On the weak point of the stronger uncertainty relation

K. Urbanowski

We analyze the uncertainty relation for the sum of variances, which is called, in some papers, the stronger uncertainty relation for all incompatible observables. We show that this uncertainty relation for the sum of variances of the observables A and B calculated for the eigenstate of one of these observables (say of B), contrary to the suggestions presented in some papers, leads to the same results as the Heisenberg–Robertson uncertainty relation; that is, it does not give any bounds on the variance of A.We analyze the uncertainty relation for the sum of variances, which is called, in some papers, the stronger uncertainty relation for all incompatible observables. We show that this uncertainty relation for the sum of variances of the observables A and B calculated for the eigenstate of one of these observables (say of B), contrary to the suggestions presented in some papers, leads to the same results as the Heisenberg–Robertson uncertainty relation; that is, it does not give any bounds on the variance of A.

arXiv:2503.09201Academia quantum.Feb 28, 20254 citations

Weighted Uncertainty Relations

Yunlong Xiao, Naihuan Jing, Xianqing Li‐Jost +1 more

Recently, Maccone and Pati have given two stronger uncertainty relations based on the sum of variances and one of them is nontrivial when the quantum state is not an eigenstate of the sum of the observables. We derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations and remove the restriction on the quantum state. Generalization to multi-observable cases is also given and an optimal lower bound for the weighted sum of the variances is obtained in general quantum situation.Recently, Maccone and Pati have given two stronger uncertainty relations based on the sum of variances and one of them is nontrivial when the quantum state is not an eigenstate of the sum of the observables. We derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations and remove the restriction on the quantum state. Generalization to multi-observable cases is also given and an optimal lower bound for the weighted sum of the variances is obtained in general quantum situation.

arXiv:1603.01004Scientific ReportsMar 17, 201642 citations

Quaternionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles

K. Mahesh Krishna

Jan 1, 20250 citations

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Comments on "Stronger Uncertainty Relations for All Incompatible Observables"

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