Some Studies On The Lower Bound Of Entropy Uncertainty Relations

Uncertainty relation is an important aspect of quantum mechanics,which is different from classical mechanics.The classical uncertainty relation uses the variance of observables as the uncertainty measure.The entropy uncertainty relation uses the entropy as the uncertainty measure.The entropy uncertainty relation has an important application in quantum information theory.Therefore,it is of great significance to study the lower bounds of the entropy uncertainty relations,especially the optimal lower bounds.Shannon entropy,Renyi entropy and Tsallis entropy are the three kinds of entropy studied most in quantum information theory,and can be unified into the so-called?h,??-entropy.In a2-dimensional Hilbert space,the optimal lower bound of the Shannon entropy uncertainty relation was obtained by the Bloch vector representation,and the optimal lower bound of the Renyi entropy uncertainty relation was obtained by the Z-Y decomposition method.Bloch vector representation and Z-Y decomposition are two different parameterization methods for a 2-dimensional quantum system,and the former represents the set of all 2-dimensional quantum states as a unit ball in the 3-dimensional Euclidean space.This paper studies the calculation of the lower bound of entropy uncertainty relations,and discusses the relationship between separability and entropy uncertainty relation.The main research contents are as follows:First of all,the calculation of the optimal lower bound of Renyi entropy uncertainty relation in a 2-dimensional Hilbert space is studied.By using the similar method of Ghirardi,for arbitrary non-negative index pair??₁,?₂?,using Bloch vector representation,the calculation of the optimal lower bound for the Renyi entropy uncertainty relation in terms of pure states is transformed into a single parameter optimization problem.In particular,an analytical solution is obtained when??₁,?₂?? [0,1/2]²,and a semianalytical solution is obtained in the case of ?₁= ?₂.In addition,when??₁,?₂?? [0,2]²,the derived lower bound is also valid for mixed states.Secondly,the calculation of the EUR lower bound for product observables in a composite system is studied.Under the framework of?h,??-entropy,using lower bounds of EURs in subsystems,the lower bounds of EUR for separable states in the composite system is derived.For Shannon entropy,we prove that Maassen-Uffink bound is "additive".We guess that,in a 2 × 2 system,the optimal Renyi EUR lower bound for product observables is "additive",that is,the sum of the optimal lower bound of the entropy uncertainty relation in the subsystem equals to the optimal global lower bound in the composite system.We give two conclusions related to the proof of this conjecture.One is to reduce the number of parameters of quantum states with the help of local unitary transformation,and the other is to transform the calculation of the optimal lower bound of Shannon EUR in the2 × 2 system into a multi-parameter optimization problem with the help of the generalized Bloch vector representation,and give a conclusion related to further simplification of the calculation.