We focus here on the uncertainty of an observable $Y$ caused by a precise measurement of $X$ . We illustrate the effect by analyzing the general scenario of two successive measurements of spin components $X$ and $Y$ . We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of $Y$ . We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.We focus here on the uncertainty of an observable $Y$ caused by a precise measurement of $X$. We illustrate the effect by analyzing the general scenario of two successive measurements of spin components $X$ and $Y$. We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of $Y$. We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.

arXiv:1401.1013Physical Review AMar 10, 201427 citations

Tighter generalized entropic uncertainty relations in multipartite systems

Lin Wu, Ye Liu, Dong Wang

The uncertainty principle, which demonstrates the intrinsic uncertainty of nature from an information-theory perspective, is at the heart of quantum information theory. In the realm of quantum information theory, Shannon entropy is used to depict the uncertainty relation in general. A tighter lower bound for uncertainty relations facilitates more accurate predictions of measurement outcomes and more robust quantum information processing. Interestingly, the tripartite entropic uncertainty relation (EUR) can be further optimized. Renes et al. proposed a tripartite EUR [J. M. Renes and J.-C. Boileau, Phys. Rev. Lett. 103, 020402 (2009)], and subsequently, Ming et al. strengthened its lower bound in [F. Ming, D. Wang, X.-G. Fan, W.-N. Shi, L. Ye, and J.-L. Chen, Phys. Rev. A 102, 012206 (2020)]. Specifically, we derive a tighter lower bound of the tripartite EUR using the Holevo quantity. Furthermore, we generalize the tripartite EUR, that is, the generalized entropic uncertainty relation for multiple measurements in multipartite systems. As illustrations, we provide several typical examples to show that our bound is tight and outperforms the previous bound. Furthermore, our findings pave the way for using the tighter bound for the quantum secret key rate in quantum key distribution protocols and are essential for quantum precision measurements in the framework of genuine multipartite systems. By providing a close peek at the nature of uncertainty, our results may find broad applications in the security analysis of quantum cryptography.The uncertainty principle, which demonstrates the intrinsic uncertainty of nature from an information-theory perspective, is at the heart of quantum information theory. In the realm of quantum information theory, Shannon entropy is used to depict the uncertainty relation in general. A tighter lower bound for uncertainty relations facilitates more accurate predictions of measurement outcomes and more robust quantum information processing. Interestingly, the tripartite entropic uncertainty relation (EUR) can be further optimized. Renes et al. proposed a tripartite EUR [J. M. Renes and J.-C. Boileau, Phys. Rev. Lett. 103, 020402 (2009)], and subsequently, Ming et al. strengthened its lower bound in [F. Ming, D. Wang, X.-G. Fan, W.-N. Shi, L. Ye, and J.-L. Chen, Phys. Rev. A 102, 012206 (2020)]. Specifically, we derive a tighter lower bound of the tripartite EUR using the Holevo quantity. Furthermore, we generalize the tripartite EUR, that is, the generalized entropic uncertainty relation for multiple measurements in multipartite systems. As illustrations, we provide several typical examples to show that our bound is tight and outperforms the previous bound. Furthermore, our findings pave the way for using the tighter bound for the quantum secret key rate in quantum key distribution protocols and are essential for quantum precision measurements in the framework of genuine multipartite systems. By providing a close peek at the nature of uncertainty, our results may find broad applications in the security analysis of quantum cryptography.

Physical Review ADec 28, 202247 citations

Entropic uncertainty relations in quantum mechanics

Iwo Białynicki‐Birula

Lecture notes in mathematicsJan 1, 198547 citations

Entropic uncertainty relations for multiple measurements assigned with biased weights

Shan Huang, Hua‐Lei Yin, Zeng‐Bing Chen +1 more

The entropic way of formulating Heisenberg's uncertainty principle not only plays a fundamental role in applications of quantum information theory but also is essential for manifesting genuine nonclassical features of quantum systems. In this paper we investigate Rényi entropic uncertainty relations (EURs) in the scenario where measurements on individual copies of a quantum system are selected with nonuniform probabilities. In contrast with EURs that characterize an observer's overall lack of information about outcomes with respect to a collection of measurements, we establish state-dependent lower bounds on the weighted sum of entropies over multiple measurements. Conventional EURs thus correspond to the special cases when all weights are equal, and in such cases we show our results are generally stronger than previous ones. Moreover, taking the entropic steering criterion as an example, we numerically verify that our EURs could be advantageous in practical quantum tasks by optimizing the weights assigned to different measurements. Importantly, this optimization does not require quantum resources and is efficiently computable on classical computers. Published by the American Physical Society 2024The entropic way of formulating Heisenberg's uncertainty principle not only plays a fundamental role in applications of quantum information theory but also is essential for manifesting genuine nonclassical features of quantum systems. In this paper we investigate Rényi entropic uncertainty relations (EURs) in the scenario where measurements on individual copies of a quantum system are selected with nonuniform probabilities. In contrast with EURs that characterize an observer's overall lack of information about outcomes with respect to a collection of measurements, we establish state-dependent lower bounds on the weighted sum of entropies over multiple measurements. Conventional EURs thus correspond to the special cases when all weights are equal, and in such cases we show our results are generally stronger than previous ones. Moreover, taking the entropic steering criterion as an example, we numerically verify that our EURs could be advantageous in practical quantum tasks by optimizing the weights assigned to different measurements. Importantly, this optimization does not require quantum resources and is efficiently computable on classical computers. Published by the American Physical Society 2024

arXiv:2309.16955Physical Review ResearchJan 31, 20249 citations

Entropic Uncertainty Relations in Quantum Physics

Iwo Białynicki‐Birula, Łukasz Rudnicki

arXiv:1001.4668Springer eBooksJan 1, 2011125 citations

We focus here on the uncertainty of an observable

Y

caused by a precise measurement of

X

. We illustrate the effect by analyzing the general scenario of two successive measurements of spin components

X

and

Y

. We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of

Y

. We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.