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熵不确定度关系综述

李丽娟 明飞 宋学科 叶柳 王栋

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熵不确定度关系综述

李丽娟, 明飞, 宋学科, 叶柳, 王栋

Review on entropic uncertainty relations

Li Li-Juan, Ming Fei, Song Xue-Ke, Ye Liu, Wang Dong
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  • 不确定关系是量子力学的基本特征之一, 随着量子信息理论的蓬勃发展, 不确定关系更是在其中发挥着重要的作用. 特别是将熵引入来描述不确定关系之后, 不确定关系在量子信息技术中涌现出多种应用. 众所周知, 熵不确定度关系已成为几乎所有量子密码协议安全分析的核心要素. 这篇综述主要回顾不确定关系的发展历史和最新研究进展, 从Heisenberg提出不相容测量其结果是不能被预测伊始, 许多学者在该观点的启发下, 做了进一步的相关扩展研究, 将可观测物体与环境之间的量子关联结合起来, 对不确定关系进行各种推广从而得到更普适的数学表达式. 除此以外, 本文还重点介绍了量子存储下的熵不确定度关系及其发展, 也介绍了在某些物理系统中对应的动力学特性. 最后讨论了熵不确定度关系在量子信息领域的各种应用, 从随机数到波粒二象性再到量子密钥分发.
    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.
      通信作者: 王栋, dwang@ahu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 12075001, 61601002, 12004006, 12175001)、安徽省自然科学基金(批准号: 1508085QF139)和中国科学院量子信息重点实验室开放基金(批准号: No. KQI201701)资助的课题
      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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  • 图 1  每个从粒子源发出的粒子都是用PQ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PQ的测量结果

    Fig. 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

    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} $$ {\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)

    Fig. 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).

    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} $$ {{R}} $ 测量, 然后向Bob告知测量选择Θ. Bob的任务是正确猜测K

    Fig. 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

    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XZ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

    Fig. 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.

    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

    Fig. 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

    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 $$ {\rm{D}}_1 $处检测到光子

    Fig. 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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  • 修回日期:  2021-12-26
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