Witness For Non-quasi Maximally Entangled States

Quantum entanglement is an important resource in quantum computing and quantum information processing tasks.Using entangled states can accomplish many tasks that can not be accomplished in classical information theory,such as quan-tum key distribution,quantum dense coding,quantum secure direct communication,quantum secret sharing and quantum teleportation,etc.The maximally entangled states,as the special entanglement,defined as those states that have maximal entan-glement of formation under some entanglement measurements,are the ideal resource for many quantum mission.In this paper,we call a convex combination of maxi-mally entangled pure states a quasi maximally entangled state.The paper mainly studied the witness for non-quasi maximally entangled states and obtained some necessary and sufficient conditions for an observable to be a witness for non-quasi maximally entangled states.The paper is divided into three chapters and arranged as follows:In chapter 1,we introduce some research background and status on our contents,and some basic definitions and theorems.In chapter 2,we introduce the definition of quasi maximally entangled states,then present the concept of the witness of non-quasi maximally entangled states,which is an observable that can distinguish non-quasi maximally entangle states from quasi maximally entangled ones.Firstly,we prove that each non-quasi maximally entangled state can be detected by a specific witness.Secondly,we obtained some necessary and sufficient conditions for an observable to be a witness for non-quasi maximally entangled states.In chapter 3,we mainly study two classes of special self-adjoint operators.The first is the tensor product operator A(?)B on the bipartite quantum system,it is proved that A(?)B can be a witness for non-quasi maximally entangled states from the theorem obtained in chapter 2.Especially,we compute non-quasi maximally entangled states that can be detected by A(?)B.The second class is the self-adjoint operator L =(A(?)B)F(A(?)B)(?),we obtain the conclusion that L can also be a witness for non-quasi maximally entangled states according to its eigenstate for the maximal eigenvalue of L is separable.