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Uncertainty is a fundamental characteristic of quantum system. The degree of uncertainty of an observable has long been investigated by the standard deviation of the observable. In recent years, however, by analyzing some special examples, researchers have found that the Shannon entropy of the measurement outcomes of an observable is more suitable to quantify its uncertainty. Formally, Shannon entropy is a special limit of a more general Rényi entropy. In this paper, we discuss the problem of how to predict the measurement outcome of an observable by the existing measurement results of the observable, and how to quantitatively describe the uncertainty of the observable from the perspective of the repeatable probability of the measurement results of this observable in an unknown state. We will argue that if the same observable of different systems in the same state is repeatedly and independently measured many times, then the probability of obtaining an identical measurement result is a decaying function of the number of measurements of obtaining the same result, and the decay rate of the repeatable probability for obtaining the same measurement results and the repeatable number of measurements can represent the degree of uncertainty of the observable in this state. It means that the greater the uncertainty of an observable, the faster the repeatable probability decays with the number of repeatable measurements; conversely, the smaller the uncertainty, the slower the repeatable probability decays with the number of repeatable measurements. This observation enables us to give the Shannon entropy and the Rényi entropy of an observable uniformly by the functional relation between the repeatable probability and the number of repeatable measurements. We show that the Shannon entropy and the Rényi entropy can be formally regarded as the “decay index” of the repeatable probability with the number of repeatable measurements. In this way we also define a generalized Rényi entropy by the repeatable probability for consecutively observing identical results of an observable, and therefore we give a proof of the Maassen-Uffink type entropic uncertainty relation by using this generalized Rényi entropy. This method of defining entropy shows that entropic uncertainty relation is a quantitative limitation for the decay rate of the total probability for obtaining identical measurement results when we simultaneously measure two observables many times.
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Keywords:
- uncertainty /
- entropy /
- uncertainty relation
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图 1 (a) 可观测量本征值的概率分布
$ {p}_{i} $ ; (b) 可观测量的重复概率$ {P}_{x}(r) $ ; (c) 重复概率$ {P}_{x}(r) $ 的导数; (d) 概率分布P的范数$||P||_{r}$ Figure 1. (a) Probability distribution
$ {p}_{i} $ of the eigenvalues of an observable; (b) repeatable probability$ {P}_{x}\left(r\right) $ of the observable; (c) derivative of the repeatable probability$ {P}_{x}\left(r\right) $ ; (d) norm$||P||_{r}$ of the probability distribution$ P $ .
图 2 (a) 概率分布
$ P $ 的Rényi熵$ {H}_{r}\left(P\right) $ ; (b)$ {H}_{1} $ ,$ {H}_{2} $ ,$ {H}_{3} $ 给出的重复概率的拟合曲线Figure 2. (a) Rényi entropy
$ {H}_{r}\left(P\right) $ of the probability distribution$ P $ ; (b) fitting curve of the repeatable probability given by$ {H}_{1} $ ,$ {H}_{2} $ ,$ {H}_{3} $ -
[1] Heisenberg W 1927 Z. Phys. 43 172 (in German)
[2] 海森堡 W (王正行, 李绍光, 张虞 译) 2017量子论的物理原理 (北京: 高等教育出版社) 第11页
Heisenberg W (translated by Wang Z X, Li S G, Zhang Y) 2017 The Physical Principles of the Quantum Theory (Beijing: Higher Education Press) p11 (in Chinese)
[3] Kennard E H 1927 Z. Phys. 44 326 (in German)
[4] Robertson H P 1929 Phys. Rev. 34 163
Google Scholar
[5] Schrödinger E 1930 Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse 14 296
[6] 曾谨言 2013 量子力学(卷I) (北京: 科学出版社) 第142页
Zeng J Y 2013 Quantum Mechanics (Vol. 1) (Beijing: Science Press) p142 (in Chinese)
[7] Everett H 1957 Rev. Mod. Phys. 29 454
Google Scholar
[8] Hirschman I I 1957 Am. J. Math. 79 152
Google Scholar
[9] Beckner W 1975 Ann. Math. 102 159
Google Scholar
[10] Białynicki-Birula I 1984 Phys. Lett. A 103 253
Google Scholar
[11] Maassen H, Uffink J B M 1988 Phys. Rev. Lett. 60 1103
Google Scholar
[12] Deutsch D 1983 Phys. Rev. Lett. 50 631
Google Scholar
[13] Coles P J, Colbeck R, Yu L, Zwolak M 2012 Phys. Rev. Lett. 108 210405
Google Scholar
[14] Berta M, Christandl M, Colbeck R, Renes J M, Renner J 2010 Nat. Phys. 6 659
Google Scholar
[15] Coles P J, Berta M, Tomamichel M, Wehner S 2017 Rev. Mod. Phys. 89 015002
Google Scholar
[16] Wang D, Ming F, Song X K, Ye L, Chen J L 2020 Eur. Phys. J. C 80 800
Google Scholar
[17] Yang Y Y, Sun W Y, Shi W N, Ming F, Wang D, Ye L 2019 Front. Phys. 14 31601
Google Scholar
[18] Chen M N, Wang D, Ye L 2019 Phys. Lett. A 383 977
Google Scholar
[19] Wu L, Ye L, Wang D 2022 Phys. Rev. A 106 062219
Google Scholar
[20] Xie B F, Ming F, Wang D, Ye L, Chen J L 2021 Phys. Rev. A 104 062204
Google Scholar
[21] Ming F, Wang D, Fan X G, Shi W N, Ye L, Chen J L 2020 Phys. Rev. A 102 012206
Google Scholar
[22] Wu L, Song X K, Ye L, Wang D 2022 AAPPS Bull. 32 24
Google Scholar
[23] 李丽娟, 明飞, 宋学科, 叶柳, 王栋 2022 物理学报 71 070302
Google Scholar
Li L J, Ming F, Song X K, Ye L, Wang D 2022 Acta. Phys. Sin. 71 070302
Google Scholar
[24] Shannon C E 1948 Bell Syst. Tech. J. 27 379
Google Scholar
[25] Rényi A 1961 Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability (Vol. 1): Contributions to the Theory of Statistics Berkeley, University of California Press, June 20–July 30, 1960, p547
[26] Stein E M 1956 Trans. Amer. Math. Soc. 83 482
Google Scholar
[27] Thorin G O 1948 Ph. D. Dissertation (Lund: Lunds University)
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