Some Characterizations Of Markov Quantum States

Markov property is a very important concept in probability theory.It means that when the current state and all past states of a random process are known,the conditional probability of the future state is only related to the current state,and has nothing to do with the state at any time in the past,so called this process has the Markov property.Markov property application is very broad.Markov model has made profound progress in language processing,such as language recognition,automatic part-of-speech tagging and phone syllable-to-character conversion.In recent years,many scholars have applied Markov properties to quantum systems and introduced Markov quantum states,making Markov quantum states play a vital role in the study of system evolution,quantum discord and quantum cryptography.In this paper,we will use operator theory and matrix theory to give the significance of studying Markov quantum state.According to the meaning of Markov quantum state and the properties of von Neumann entropy,the pure state and mixed state of Markov quantum state are studied respectively.This paper is divided into four chapters:In Chapter 1,it introduces the research background,current situation and relevant preparatory knowledge of this paper.In Chapter 2,the conditional mutual information expression and related properties are firstly given in classical system.And then generalize to quantum system,the conditional mutual information of three-body quantum state is given,and several properties of conditional mutual information are deduced by using t he von Neumann entropy correlation theorem.Finally,the definition of Markov quantum state is given.In Chapter 3,based on the von Neumann entropy property of pure states,we discuss the relationship between Markov pure states and separable states,and give examples to apply the theorem.Secondly,the form of the three-body state with Schdimt decomposition is obtained when it is a Markov quantum state.In Chapter 4,from the point of view of mixed states,it is firstly discussed that when the three-body quantum state is the product state,the quantum state should meet some conditions for the Markov quantum state.Secondly,several special Markov quantum mixed states are given,and then the relationship between Markov quantum state and classical correlation state,complete classical correlation state,double classical correlation state and single classical correlation state is discussed,and some examples are given to apply the theorem.Finally,it is proved that three special quantum operations can preserve the Markov properties of quantum states.