Study On The Uncertainty Relation And Optimization Of Unitary Operator

As the foundation of quantum mechanics,uncertainty principle is an important research topic in quantum information theory.It reflects the basic law of the motion of microscopic particles,that is,it is impossible to simultaneously and accurately measure two observables that are incompatible,such as the position and momentum of a particle.Entropy is one of the important concepts in quantum information theory.Entropy of quantum system is a measure of system randomness.Entropy also quantifies how much information we gain about a system on an average,or to quantify the resources needed to store information.Uncertainty principle and quantum entropy are widely used in quantum entanglement.Following these two directions,this paper is devoted to the study of quantum uncertainty relations and entropy-related bound.The main research contents are as follows.We derive the lower bound of uncertainty relations of two unitary operators for a class of states based on the geometric-arithmetic inequality and Cauchy–Schwarz inequality.Furthermore,we improve the decreasing sequence of two unitary operators to be finer.Finally,we establish the decreasing sequence of the uncertainty relation of three unitary operators,and obtain the uncertainty relation of the optimized three unitary operators.Compared to the known bound introduced in Yu et al(2019 Phys.Rev.A 100 022116),the unitary uncertainty relations bound with our method is tighter,to a certain extent.Meanwhile,some examples are given in the paper to illustrate our conclusions.Based on the Rényi entropy and Tsallis entropy,at first,we derive the bounds of the expectation value and variance of quantum observable respectively.By the maximal value of Rényi entropy,we show an upper bound on the product of variance and entropy.Furthermore,we obtain the reverse uncertainty relation for the product and sum of the variances for n observables respectively.