Research On Some Problems About LOCC

In quantum information theory, local operation with classical communication (which is short for’LOCC) plays a central role. In this paper, we study some problems about LOCC. In the first part, we study the question of discrimination of quantum states which are mutually orthogonal maximally entangled. Given a state from a known set of quantum states, our goal is to determine the state using a measurement on entire composite system. If the states are mutually orthogonal, then they can be perfectly distinguished; if they are linearly independent, then they can be distinguished with a certain probability. In the bipartite quantum system, two parties are separated, so they can only perform local operations on the subsystem, and change the classical information. In some case, it is possible to identify exactly the state by LOCC, while in some others, it is not. In the second part, the definition of quantum instruments (QI) and the relationships among different classes of QIs have been proposed by Eric Chitambar, but there are not detail introductions about bipartite states, which are very important in the study of local operations with classical communication. In this part, we discuss the structures about LOCC and the special relationships among different classes of QIs on the composite system H=HA(?)HB.In the third part, separability of quantum state and partial positive transpose in finite dimensional tensor product spaces are discussed. With operator theory, we introduce the separable operators, PPT operators, and discuss the properties of these operators on finite dimensional tensor product space. Lastly, separable operators and PPT operators, separable maps and PPT maps are discussed. This paper is divided into three chapters.In Chapter1, firstly, some basic concepts such as LOCC measurement, perfect LOCC discrimination and maximally entangled state are introduced. Secondly, we discuss sufficient conditions and necessary conditions for perfect LOCC discrimi-nation, and present the relationship between the perfect LOCC discrimination of orthogonal maximally entangled states and the discrimination of corresponding u-nitary operators. In Chapter2, we study the structure of LOCC. We first introduce the concept of quantum instrument which is founded by Eric Chitambar. Next, we discuss the relationships among the different classes of QIs on the composite system.In Chapter3, firstly, we extend the separable state and PPT to the operator theory, and give the concepts of separable operator and PPT operator. Secondly, we discuss some properties about separable operators and PPT operators.