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Entirety of Quantum Uncertainty and Its Experimental Verification

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【Author】 Jie Xie;Li Zhou;Aonan Zhang;Huichao Xu;Man-Hong Yung;Ping Xu;Nengkun Yu;Lijian Zhang;National Laboratory of Solid State Microstructures,College of Engineering and Applied Sciences and School of Physics,Nanjing University;Collaborative Innovation Center of Advanced Microstructures,Nanjing University;Department of Computer Science and Technology,Tsinghua University;Shenzhen Institute for Quantum Science and Engineering and Department of Physics,Southern University of Science and Technology;Shenzhen Key Laboratory of Quantum Science and Engineering,Southern University of Science and Technology;Institute for Quantum Information & State Key Laboratory of High Performance Computing,College of Computer,National University of Defense Technology;Centre for Quantum Software and Information,School of Software,Faculty of Engineering and Information Technology,University of Technology Sydney;

【通讯作者】俞能昆;张利剑;

【机构】 National Laboratory of Solid State Microstructures,College of Engineering and Applied Sciences and School of Physics,Nanjing University； Collaborative Innovation Center of Advanced Microstructures,Nanjing University； Department of Computer Science and Technology,Tsinghua University； Shenzhen Institute for Quantum Science and Engineering and Department of Physics,Southern University of Science and Technology； Shenzhen Key Laboratory of Quantum Science and Engineering,Southern University of Science and Technology； Institute for Quantum Information & State Key Laboratory of High Performance Computing,College of Computer,National University of Defense Technology； Centre for Quantum Software and Information,School of Software,Faculty of Engineering and Information Technology,University of Technology Sydney；

【摘要】 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.更多还原

【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.更多还原

【关键词】 uncertainty； formulated； necessarily； arbitrary； projective； exactly； photonic； polynomial； union； constraint；

【基金】 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

【文献出处】 Chinese Physics Letters ,中国物理快报(英文版) , 编辑部邮箱 ,2021年07期

【分类号】O413
【下载频次】20

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