Recently, an entropic uncertainty relation for multiple measurements was proposed by Liu et al. [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements. Herein, we improve the lower bound of the entropic uncertainty relation for multiple measurements, referred to as a simply constructed bound (SCB). We verify that the SCB is tighter than Liu et al.'s result for arbitrary mutually unbiased basis measurements, which might play a fundamental and crucial role in practical quantum information processing. Moreover, we optimize the SCB by considering mutual information and the Holevo quantity, and propose an optimized SCB. Notably, the proposed bounds are extrapolations of the behavior of two measurements to a larger collection of measurements. It is believed that our findings would shed light on entropy-based uncertainty relations in the multiple measurement scenario and will be beneficial for security analysis in quantum key distributions.
Recently, an entropic uncertainty relation for multiple measurements was proposed by Liu et al. [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements. Herein, we improve the lower bound of the entropic uncertainty relation for multiple measurements, referred to as a simply constructed bound (SCB). We verify that the SCB is tighter than Liu et al.'s result for arbitrary mutually unbiased basis measurements, which might play a fundamental and crucial role in practical quantum information processing. Moreover, we optimize the SCB by considering mutual information and the Holevo quantity, and propose an optimized SCB. Notably, the proposed bounds are extrapolations of the behavior of two measurements to a larger collection of measurements. It is believed that our findings would shed light on entropy-based uncertainty relations in the multiple measurement scenario and will be beneficial for security analysis in quantum key distributions.
Recently, an entropic uncertainty relation for multi-measurement has been proposed by Liu et al. in [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements. Herein, we improve the lower bound of the entropic uncertainty relation for multi-measurement, termed as simply constructed bound (SCB). We verify that the SCB is tighter than Liu et al.'s result for arbitrary mutually unbiased basis measurements, which might play a fundamental and crucial role in practical quantum information processing. Moreover, we optimize the SCB by considering mutual information and the Holevo quantity, and propose an optimized SCB (OSCB). Notably, the proposed bounds are extrapolations of the behavior of two measurements to a larger collection of measurements. It is believed that our findings would shed light on entropy-based uncertainty relations in the multi-measurement scenario and will be beneficial for security analysis in quantum key distributions.
Entropic uncertainty relations demonstrate the intrinsic uncertainty of nature from an information-theory perspective. Recently, a quantum-memory-assisted entropic uncertainty relation for multiple measurements was proposed by Wu et al. [Phys. Rev. A 106, 062219 (2022)]. Interestingly, the quantum-memory-assisted entropic uncertainty relation for multiple measurement settings can be further generalized. In this work, we propose two complementary multipartite quantum-memory-assisted entropic uncertainty relations and our lower bounds depend on values of complementarity of the observables, (conditional) von Neumann entropies, Holevo quantities, and mutual information. As an illustration, we provide several typical cases to exhibit that our bounds are tighter and outperform the previous bounds.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
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 is a striking and fundamental feature in quantum mechanics, distinguishing it from classical mechanics. It offers an important lower bound to predict outcomes of two arbitrary incompatible observables measured on a particle. In quantum information theory, this uncertainty principle is popularly formulized in terms of entropy. Here, we present an improvement of the tripartite quantum-memory-assisted entropic uncertainty relation. The uncertainty's lower bound is derived by considering mutual information and the Holevo quantity. It shows that the bound derived by this method will be tighter than the lower bound of Renes and Boileau [Phys. Rev. Lett. 103, 020402 (2009)]. Furthermore, regarding a pair of mutual unbiased bases as the incompatibility, our bound will become extremely tight for the three-qubit X-state system, completely coinciding with the entropy-based uncertainty, and can restore Renes and Boileau's bound with respect to arbitrary tripartite pure states. In addition, by applying our lower bound one can attain the tighter bound of the quantum secret key rate, which is of basic importance to enhance the security of quantum key distribution protocols.
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 2024