| GENERAL |
|
|
|
|
Entirety of Quantum Uncertainty and Its Experimental Verification |
| Jie Xie1,2†, Li Zhou3†, Aonan Zhang1,2, Huichao Xu1,2, Man-Hong Yung4,5, Ping Xu1,2,6, Nengkun Yu7*, and Lijian Zhang1,2* |
1National Laboratory of Solid State Microstructures, College of Engineering and Applied Sciences and School of Physics, Nanjing University, Nanjing 210093, China
2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
3Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
4Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
5Shenzhen Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
6Institute for Quantum Information & State Key Laboratory of High Performance Computing, College of Computer, National University of Defense Technology, Changsha 410073, China
7Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW, Australia
|
|
| Cite this article: |
|
Jie Xie, Li Zhou, Aonan Zhang et al 2021 Chin. Phys. Lett. 38 070303 |
|
|
|
|
Abstract As a foundation of quantum physics, uncertainty relations describe ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertainty relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position and momentum operators, Heisenberg's uncertainty principle points exactly to the attainable area of the variances of position and momentum. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of (a) a pair of one-qubit observables and (b) a pair of projective observables for arbitrary dimension, and give the first experimental observation of such areas in a photonic system.
|
|
|
Received: 07 March 2021
Published: 21 June 2021
|
|
| PACS: |
03.65.Ta
|
(Foundations of quantum mechanics; measurement theory)
|
| |
42.50.Xa
|
(Optical tests of quantum theory)
|
|
|
| Fund: Supported by the National Key Research and Development Program of China (Grant No. 2017YFA0303703), the National Natural Science Foundation of China (Grant Nos. 91836303, 61975077, 61490711, 11690032, 11875160, and U1801661), the Natural Science Foundation of Guangdong Province (Grant No. 2017B030308003), the Key R&D Program of Guangdong Province (Grant No. 2018B030326001), the Science, Technology and Innovation Commission of Shenzhen Municipality (Grant Nos. JCYJ20170412152620376, JCYJ20170817105046702, and KYTDPT20181011104202253), the Economy, Trade and Information Commission of Shenzhen Municipality (Grant No. 201901161512), Guangdong Provincial Key Laboratory (Grant No. 2019B121203002), ARC DECRA 180100156 and ARC DP210102449. |
|
|
|
| [1] | Heisenberg W 1927 Z. Phys. 43 172 |
| [2] | Gühne O 2004 Phys. Rev. Lett. 92 117903 |
| [3] | Berta M, Christandl M, Colbeck R et al. 2010 Nat. Phys. 6 659 |
| [4] | Li C F, Xu J S, Xu X Y et al. 2011 Nat. Phys. 7 752 |
| [5] | Prevedel R, Hamel D R, Colbeck R et al. 2011 Nat. Phys. 7 757 |
| [6] | Oppenheim J and Wehner S 2010 Science 330 1072 |
| [7] | Kaniewski J M K, Tomamichel M, and Wehner S 2014 Phys. Rev. A 90 012332 |
| [8] | Busch P, Lahti P, and Werner R F 2014 Rev. Mod. Phys. 86 1261 |
| [9] | Coles P J, Berta M, Tomamichel M et al. 2017 Rev. Mod. Phys. 89 015002 |
| [10] | Ozawa M 2003 Phys. Rev. A 67 042105 |
| [11] | Busch P, Lahti P, and Werner R F 2013 Phys. Rev. Lett. 111 160405 |
| [12] | Buscemi F, Hall M J W, Ozawa M et al. 2014 Phys. Rev. Lett. 112 050401 |
| [13] | Zhou F, Yan L, Gong S et al. 2016 Sci. Adv. 2 e1600578 |
| [14] | Kennard E H 1927 Z. Phys. 44 326 |
| [15] | Weyl H 1927 Z. Phys. 46 1 |
| [16] | Robertson H P 1929 Phys. Rev. 34 163 |
| [17] | Schrödinger E 1930 Proc. Prussian Acad. Sci. 19 296 |
| [18] | Maccone L and Pati A K 2014 Phys. Rev. Lett. 113 260401 |
| [19] | Pati A and Sahu P 2007 Phys. Lett. A 367 177 |
| [20] | Huang Y 2012 Phys. Rev. A 86 024101 |
| [21] | Chen B and Fei S M 2015 Sci. Rep. 5 14238 |
| [22] | Chen B, Cao N P, Fei S M et al. 2016 Quantum Inf. Process. 15 3909 |
| [23] | Mondal D, Bagchi S, and Pati A K 2017 Phys. Rev. A 95 052117 |
| [24] | de Guise H, Maccone L, Sanders B C et al. 2018 Phys. Rev. A 98 042121 |
| [25] | Hirschman I I 1957 Am. J. Math. 79 152 |
| [26] | Beckner W 1975 Ann. Math. 102 159 |
| [27] | Deutsch D 1983 Phys. Rev. Lett. 50 631 |
| [28] | Maassen H and Uffink J B M 1988 Phys. Rev. Lett. 60 1103 |
| [29] | Sánches-Ruiz J 1998 Phys. Lett. A 244 189 |
| [30] | Coles P J and Piani M 2014 Phys. Rev. A 89 022112 |
| [31] | Xiao Y, Jing N, and Li-Jost X 2017 Quantum Inf. Process. 16 104 |
| [32] | Demirel B, Sponar S, Abbott A A et al. 2019 New J. Phys. 21 013038 |
| [33] | Abbott A A, Alzieu P L, Hall M J W et al. 2016 Mathematics 4 1 |
| [34] | Schwonnek R, Dammeier L, and Werner R F 2017 Phys. Rev. Lett. 119 170404 |
| [35] | Sponar S, Danner A, Obigane K et al. 2020 Phys. Rev. A 102 042204 |
| [36] | Zhao Y Y, Xiang G Y, Hu X M et al. 2019 Phys. Rev. Lett. 122 220401 |
| [37] | Li J L and Qiao C F 2015 Sci. Rep. 5 12708 |
| [38] | Abbott A, Alzieu P L, Hall M et al. 2016 Mathematics 4 8 |
| [39] | Busch P and Reardon-Smith O 2019 arXiv:1901.03695v1 [quant-ph] |
| [40] | Coste M 2002 An Introduction to Semialgebraic Geometry (Citeseer) |
| [41] | Böttcher A and Spitkovsky I 2010 Linear Algebra Its Appl. 432 1412 |
| [42] | Reck M, Zeilinger A, Bernstein H J et al. 1994 Phys. Rev. Lett. 73 58 |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
