Optimized entropic uncertainty relations for multiple measurements

Bo-Fu Xie, Fei Ming, Dong Wang, Liu Ye, and Jing-Ling Chen

Phys. Rev. A 104, 062204 – Published 10 December 2021

Abstract

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.

Received 1 June 2021
Revised 15 November 2021
Accepted 26 November 2021

DOI:https://doi.org/10.1103/PhysRevA.104.062204

Physics Subject Headings (PhySH)

Quantum foundations Quantum measurements

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Bo-Fu Xie¹, Fei Ming¹, Dong Wang ^1,2,*, Liu Ye¹, and Jing-Ling Chen^3,†

¹School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, People's Republic of China
²CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
³Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People's Republic of China

^*dwang@ahu.edu.cn
^†chenjl@nankai.edu.cn

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Vol. 104, Iss. 6 — December 2021

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Images

Figure 1
Uncertainty and the lower bounds vs Werner state's purity $p$ . The red solid line represents the entropic uncertainty [the left-hand side of Eq. (6)], the blue dashed line shows our SCB [the right-hand side of Eq. (6)], and the black dotted-dashed line shows Liu et al.'s result [the right-hand side of Eq. (5)]. The green triangle represents our OSCB [the right-hand side of Eq. (12)].
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Figure 2
Uncertainty and the lower bounds vs the Bell-diagonal state's purity $p$ . The red solid line represents the entropic uncertainty [the left-hand side of Eq. (6)], the blue dashed line shows our SCB [the right-hand side of Eq. (6)], and the black dotted-dashed line shows Liu et al.'s result [the right-hand side of Eq. (5)]. The green triangle represents our OSCB [the right-hand side of Eq. (12)].
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Figure 3
The entropic uncertainty and three lower bounds under MUB multiple measurements (three two-dimensional Pauli operators ${\hat{σ}}_{x}, {\hat{σ}}_{y}, {\hat{σ}}_{z}$ ) for $10^{6}$ randomly generated two-qubit states. (a) The $x$ axis represents our simply constructed bound (SCB) in Eq. (6) and the $y$ axis denotes the entropic uncertainty in Eq. (6). The red dashed line is the proportional function with a slope of unity, showing that the uncertainty is equal to SCB. (b) The orange points stand for Liu et al.'s bound (LMF) in Eq. (5), the purple points denote our proposed bound (SCB) in Eq. (6), and $S (A | B)$ is the conditional von Neumann entropy of state $ρ_{A B}$ . (c) The $x$ axis represents our optimized simply constructed bound (OSCB) in Eq. (12), the $y$ axis represents the entropic uncertainty in Eq. (6), and the blue dashed line is the proportional function with a slope of unity, showing that the uncertainty is equal to OSCB. (d) SCB represents the lower bound in Eq. (6) and OSCB denotes the lower bound in Eq. (12). The red dashed line is the proportional function with a slope of unity, showing that OSCB is equal to SCB in this case.
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Figure 4
The entropic uncertainty and three lower bounds under MUB multiple measurements (three three-dimensional operators $\hat{α}, \hat{β}, \hat{γ}$ ) for $10^{6}$ randomly generated two-qubit states. (a) The $x$ axis represents our simply constructed bound (SCB) in Eq. (6) and the $y$ axis denotes the entropic uncertainty in Eq. (6). The red dashed line is the proportional function with a slope of unity, displaying that the uncertainty is equal to SCB. (b) The orange points stand for Liu et al.'s bound (LMF) in Eq. (5) and the purple points denote our proposed our proposed bound (SCB), and $S (A | B)$ is the conditional von Neumann entropy of state $ρ_{A B}$ . (c) The $x$ axis represents our optimized simply constructed bound (OSCB) in Eq. (12) and the $y$ axis represents the entropic uncertainty in Eq. (6), and the blue dashed line is the proportional function with a slope of unity, displaying that the uncertainty is equal to OSCB. (d) SCB represents the lower bound in Eq. (6) and OSCB denotes the lower bound in Eq. (12). The red dashed line is the proportional function with a slope of unity, displaying that OSCB is equal to SCB.
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