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.
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.
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.
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 are powerful tools, especially in quantum cryptography. They typically bound the amount of uncertainty a third-party adversary may hold on a measurement outcome as a result of the measurement overlap. However, when the two measurement bases are biased towards one another, standard entropic uncertainty relations do not always provide optimal lower bounds on the entropy. Here, we derive a new entropic uncertainty relation, for certain quantum states, which can provide a significantly higher bound even if the two measurement bases are no longer mutually unbiased. We evaluate our bound on two different quantum cryptographic protocols, including BB84 with faulty/biased measurement devices, and show that our new bound can produce substantially higher key-rates under several scenarios when compared with prior work using standard entropic uncertainty relations.
Quantum uncertainty relations impose fundamental limits on the joint knowledge that can be acquired from complementary observables: Perfect knowledge of a quantum state in one basis implies maximal indetermination in all other mutually unbiased bases (MUBs). Uncertainty relations derived from joint properties of the MUBs are generally assumed to be uniform, irrespective of the specific observables chosen within a set. In this work, we demonstrate instead that the uncertainty relations can depend on the choice of observables. Through both experimental observation and numerical methods, we show that selecting different sets of three MUBs in a five-dimensional quantum system results in distinct uncertainty bounds, i.e., in varying degrees of complementarity, in terms of both entropy and variance.
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.