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 fo...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.展开更多
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)+5 种基金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.
文摘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.
