Review on entropic uncertainty relations

Li Li-Juan; Ming Fei; Song Xue-Ke; Ye Liu; Wang Dong

doi:10.7498/aps.71.20212197

Abstract

The Heisenberg uncertainty principle is one of the characteristics of quantum mechanics. With the vigorous development of quantum information theory, uncertain relations have gradually played an important role in it. In particular, in order to solved the shortcomings of the concept in the initial formulation of the uncertainty principle, we brought entropy into the uncertainty relation, after that, the entropic uncertainty relation has exploited the advantages to the full in various applications. As we all know the entropic uncertainty relation has became the core element of the security analysis of almost all quantum cryptographic protocols. This review mainly introduces development history and latest progress of uncertain relations. After Heisenberg's argument that incompatible measurement results are impossible to predict, many scholars, inspired by this viewpoint, have made further relevant investigations. They combined the quantum correlation between the observable object and its environment, and carried out various generalizations of the uncertainty relation to obtain more general formulas. In addition, it also focuses on the entropy uncertainty relationship and quantum-memory-assisted entropic uncertainty relation, and the dynamic characteristics of uncertainty in some physical systems. Finally, various applications of the entropy uncertainty relationship in the field of quantum information are discussed, from randomnesss to wave-particle duality to quantum key distribution.

Keywords:

Authors and contacts

1.
School of Physics and Optoelectronics Engineering, Anhui University, Hefei 230601, China

Corresponding author: Wang Dong, dwang@ahu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12075001, 61601002, 12004006, 12175001), the Natural Science Foundation of Anhui Province, China (Grant No. 1508085QF139), and the Fund from CAS Key Laboratory of Quantum Information (Grant No. KQI201701)

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Cited By

图 1 每个从粒子源发出的粒子都是用P或Q来测量的, 测量的选择是随机的. 不确定关系指出我们不能预测P和Q的测量结果

Figure 1. Each particle emitted from the particle source is measured by P or Q, and the choice of measurement is random. The uncertainty relation indicates that we cannot predict the outcomes of both P and Q

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图 2 玩家Alice和Bob的猜测游戏. 首先, Bob准备$ \rho_A $并把A发送给Alice. 然后, Alice以相等的概率进行$ \mathbb{Q} $或$ {\mathbb{R}} $测量, 并将测量选项存储在Θ中. 第三, Alice得出测量结果并将其存储在K, 且向Bob透露测量选择Θ. Bob的任务是猜测K (给定Θ)

Figure 2. A guessing game between players Alice and Bob. First, Bob prepares $ \rho_A $ and sends A to Alice. Then, Alice performs measurement $ \mathbb{Q} $ or $ {\mathbb{R}} $ with equal probability on A, and stores the measurement options in Θ. Third, Alice stores the measurement result in the K bit and tells Bob about her option Θ. Bob’s task is to guess K (given Θ).

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图 3 量子存储下的不确定游戏. 首先, Bob准备态${\boldsymbol{\rho}} _{AB}$, 然后把子系统A发送给Alice. 第二, Alice对A进行${Q} $和$ {{R}} $ 测量, 然后向Bob告知测量选择Θ. Bob的任务是正确猜测K

Figure 3. The guessing game with a quantum memory system. First, Bob prepares $ \rho_{AB} $ and sends A to Alice, Then, Alice performs measurement $ {Q} $ or $ {{R}} $ on A, and stores the measurement options in Θ. Third, Alice tells Bob about her option Θ. Bob’s task is to guess K correctly

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图 4 三粒子量子存储器设置图. 首先, 粒子源准备$ {\boldsymbol{\rho}} _{ABC} $, 并将A发送给Alice, B发送给Bob, C给Charlie. 接着, Alice在A上进行X或Z测量, 然后在已经给Bob粒子B的情况下, 询问Bob关于Alice的X测量结果的不确定性, 在已经给Charlie粒子C的情况下询问Charlie有关Alice的Z测量结果的不确定性. 只有他们两个同时猜出结果K这个游戏才能算Bob和Charlie胜利

Figure 4. The tripartite quantum memory setup. First, the particle source prepares $ {\boldsymbol{\rho }}_{ABC} $, and sends A to Alice, B to Bob, and C to Charlie. Next, Alice performs measurement X or Z on A, and asks Bob about the uncertainty of Alice’s X measurement outcome, ask Charlie about the uncertainty of Alice’s Z measurement outcome. Only both of them guessed that the output is K, the game can be considered a victory for Bob and Charlie.

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图 5 这两张图引用自参考文献[44] 中的第三, 四幅图, 图片展示了Ming等的结果(图上的Ref. [45]就是本文参考文献[43])和Dolatkhah等结果的对比, 这里选取的测量是泡利测量:$ X = {\sigma _x}, Z = {\sigma _z} $. 图中蓝线是式(61)左式, 红线对应右式, 重合表明对应的量子态、界与不确定度重合. (a) 广义W态量子存储下的熵不确定度及下界的图像. (b)混合三比特态量子存储下的熵不确定度及下界的图像

Figure 5. These two pictures are quoted in the third and fourth pictures in the reference [44].The picture shows the comparison of the results of Ming et al. (Ref. [45] on the picture is the reference [43] in this text) and Dolatkhah et al.. The measurement selected here is the Pauli measurement: $ X = {\sigma _x}, Z = {\sigma _z} $. The blue line in the figure is the left side of the formula (61), and the red line corresponds to the right side. Their overlap indicates the corresponding quantum state, and the bounds coincide with the uncertainty. (a) Different lower bounds of the tripartite quantum-memory-assisted entropic uncertainty relation (QMA-EUR) for the generalized W state; (b) Different lower bounds of the tripartite QMA-EUR for symmetric family of mixed three-qubit states

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图 6 这张图引用自参考文献[105]中的第18幅图, 展示的是一个Mach-Zehnder单光子干涉仪. 一个光子撞击分束器, 然后通过$ {{Z}} $的基态$ | 0 \rangle, | 1 \rangle $标记这两个可能的路径, 光子可能与干涉仪内部的某个环境E相互作用. 然后将一个相位ϕ应用于下路径, 再将这两个路径在第二个波束分束器上重新组合. 最后在$ {\rm{D}}_0 $或$ {\rm{D}}_1 $处检测到光子

Figure 6. This picture is from the 18 th picture in the reference [105]. The picture shows a Mach-Zehnder single photon interferometer. A photon hits the beam splitter, and then we pass the ground state of $ {{Z}} $ ($ | 0 \rangle, | 1 \rangle $) to mark these two possible paths. The photon may be related to an environment in the interferometer $ E $ Interaction. Then apply a phase ϕ to the lower path, and then recombine the two paths on the second beam splitter. Finally, a photon is detected at $ {\rm{D}}_0 $ or $ {\rm{D}}_1 $

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map

[1]	Heisenberg W 1927 Z. Phys. 43 172 Google Scholar
[2]	Kennard E H 1927 Z. Phys. 44 326 Google Scholar
[3]	Robertson H P 1929 Phys. Rev. 34 163 Google Scholar
[4]	Deutsch D 1983 Phys. Rev. Lett. 50 631 Google Scholar
[5]	Everett H 1957 Rev. Mod. Phys. 29 454 Google Scholar
[6]	Hirschman I I 1957 Am. J. Math. 79 152 Google Scholar
[7]	Kraus K 1987 Phys. Rev. D 35 3070 Google Scholar
[8]	Maassen H, Uffink J 1988 Phys. Rev. Lett. 60 1103 Google Scholar
[9]	Berta M, Christandl M, Colbeck R, Renes J M, Renner R 2010 Nat. Phys. 6 659 Google Scholar
[10]	Renes J, Boileau J C 2009 Phys. Rev. Lett. 103 020402 Google Scholar
[11]	Schrödinger E 1930 Physikalisch-Mathematische Klasse 14 296
[12]	Maccone L, Pati A K 2014 Phys. Rev. Lett. 113 260401 Google Scholar
[13]	Wang K K, Zhan X, Bian Z H, Li J, Zhang Y S, Xue P 2016 Phys. Rev. A 93 052108 Google Scholar
[14]	Xiao L, Wang K, Zhan X, Bian Z, Li J, Zhang Y, Xue P, Pati A K 2017 Opt. Express 25 17904 Google Scholar
[15]	Fan B, Wang K K, Xiao L, Xue P 2018 Phys. Rev. A 98 032118 Google Scholar
[16]	Białynicki-Birula I, Mycielski J 1975 Commun. Math. Phys. 44 129 Google Scholar
[17]	Shannon C 1948 Bell Syst. Tech. J. 27 379 Google Scholar
[18]	Korzekwa K, Lostaglio M, Jennings D, Rudolph T 2014 Phys. Rev. A 89 042122 Google Scholar
[19]	Rényi A 1961 Proceedings of the 4th Berkeley Symposiumon Mathematical Statistics and Probability (Vol. 1) (Berkeley: University of California Press) pp547–561
[20]	Dodonov V V, Dodonov A V 2015 Phys. Scr. 90 074049 Google Scholar
[21]	Rastegin A E 2019 Ann. Phys. 531 1800466 Google Scholar
[22]	Pegg D T 1998 Phys. Rev. A 58 4307 Google Scholar
[23]	Partovi M H 2011 Phys. Rev. A 84 052117 Google Scholar
[24]	Friedland S, Gheorghiu V, Gour G 2013 Phys. Rev. Lett. 111 230401 Google Scholar
[25]	Puchała Z, Rudnicki Ł, Życzkowski K 2013 J. Phys. A 46 272002 Google Scholar
[26]	Nielsen M A, Chuang I L (translated by Zheng D Z and Zhao Q C) 2005 Quantum Computation and Quantum Information (Beijing: Tsinghua University Press) pp155–157
[27]	Li C F, Xu J S, Xu X Y, Li K, Guo G C 2011 Nat. Phys. 7 752 Google Scholar
[28]	Prevedel R, Hamel D R, Colbeck R, Fisher K, Resch K J 2011 Nat. Phys. 7 757 Google Scholar
[29]	Xu Z Y, Zhu S Q, Yang W L 2012 Appl. Phys. Lett. 101 244105 Google Scholar
[30]	Pati A K, Wilde M M, Usha Devi A R, Rajagopal A K, Sudha 2012 Phys. Rev. A 86 042105 Google Scholar
[31]	Ollivier H, Zurek W H 2001 Phys. Rev. Lett. 88 017901 Google Scholar
[32]	Hu M L, Fan H 2013 Phys. Rev. A 88 014105 Google Scholar
[33]	Bera M N, Prabhu R, Sen (De) A, Sen U 2012 Phys. Rev. A 86 012319 Google Scholar
[34]	Coles P J, Piani M 2014 Phys. Rev. A 89 022112 Google Scholar
[35]	Adabi F, Salimi S, Haseli S 2016 Phys. Rev. A 93 062123 Google Scholar
[36]	Haseli S, Ahmadi F 2019 Eur. Phys. J. D 73 65 Google Scholar
[37]	Xie B F, Ming F, Wang D, Ye L, Chen J L 2021 Phys. Rev. A 104 062204 Google Scholar
[38]	Liu S, Mu L Z, Fan H 2015 Phys. Rev. A 91 042133 Google Scholar
[39]	Zhang J, Zhang Y, Yu C S 2015 Sci. Rep. 5 11701 Google Scholar
[40]	Dolatkhah H, Haseli S, Salimi S, Khorashad A S 2019 Quantum Inf. Process. 18 13 Google Scholar
[41]	Hu M L, Fan H 2013 Phys. Rev. A 87 022314 Google Scholar
[42]	Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[43]	Ming F, Wang D, Fan X G, Shi W N, Ye L, Chen J L 2020 Phys. Rev. A 102 012206 Google Scholar
[44]	Dolatkhah H, Haseli S, Salimi S, Khorashad A S 2020 Phys. Rev. A 102 052227 Google Scholar
[45]	Yao Y B, Wang D, Ming F, Ye L 2020 J. Phys. B: At. Mol. Opt. Phys. 53 035501 Google Scholar
[46]	Wang D, Ming F, Huang A J, Sun W Y, Shi J D, Ye L 2017 Sci. Rep. 7 1066 Google Scholar
[47]	Wang D, Shi W N, Ming F, Hoehn R D, Sun W Y, Ye L, Kais S 2018 Quantum Inf. Process. 17 335 Google Scholar
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