Entropic Uncertainty Relations And Its Meaning

The most profound difference between quantum mechanics and classical mechanics is that the former always has inherent randomness or uncertainty,that is,it usually can only give probabilistic rather than deterministic predictions about observed phenomena.Therefore,how to quantitatively describe the degree of uncertainty is undoubtedly very important in quantum mechanics.The traditional way of quantifying this uncertainty is by the standard deviations of the measurement results of observables.However,through in-depth analysis of some specific examples,it is found that the standard deviation is not the most suitable quantity to quantitatively describe the uncertainty.In recent years,some research results have shown that a more appropriate quantity to quantify uncertainty of an observable is the entropy of the probability distribution of this observable,and the traditional uncertainty relation represented by the standard deviations is also modified into an uncertainty relation represented by the entropies of observables.This entropic uncertainty relation is concerned with the probabilities of various observables and has nothing to do with the eigenvalues of these observables.It requires that the total entropy of probability distributions of the system under different bases is greater than a positive value that does not depend on specific quantum state.In this thesis,we first derive the expressions of Rényi entropy and Shannon entropy,which are now used to describe the uncertainty of an observable,from the repeatable probability of multiple occurrence of the same measurement result for the observable.Then a complete proof of the uncertainty relation represented by the entropy of the observables is given.The specific manifestations of entropy uncertainty relations in several examples and their physical meaning reflected in these examples are also discussed.The structure of this thesis is as follows:The first chapter briefly reviews the origin,evolution and history of theoretical proofs of the traditional uncertainty relations given in the usual quantum mechanics textbooks.The differences and connections among several uncertain relations in history,which are slightly different in terms of their physical ideas,expressions and methods of proofs,are also discussed.In second chapter,we first analyze the problems existing in expressing uncertainty by standard deviation,and the defects of the uncertainty relation based on standard deviation.Then we discuss the underlying basis why entropy can be used to describe uncertainty,and give a simple proof of entropy uncertainty relation.Some problems existing in expressing uncertainty with entropy are also discussed.Chapter 3 discusses the application of the entropic uncertainty relation in several concrete examples.The first example considers that when two different spin components of a spin 1/2particle are measured separately,the relation between the minimum total entropy of the measurement result,given by the entropic uncertainty relation,and the directions of two measured spin components.Another example is a system composed of two spin 1/2 particles.When measuring its probabilities in two different spin polarization configurations,we discuss the relation between the minimum entropy of the measurement results and the two sets of the directions of spin components,as well as the lower bound of the total entropy of probability distributions for two spins in a specific spin-polarized state and in a maximum entangled state,respectively.The fourth chapter summarizes the main conclusions of this paper and its significance,looks forward to the possible impact of the development of entropic uncertainty relation on the foundation of quantum mechanics and its possible application in quantum information technology.