Rsearches On Local Distinguishability Of Generalized Bell States

The local distinguishability of orthogonal quantum states is one of the fundamental problems of quantum information,it is not only an effective method to study the special relationship between quantum entanglement and quantum non-locality,but also has an important application in the design of quantum cryptographic protocols,which is a research hot spot in the field of quantum information.In this paper the one way Local Operations and Classical Communication(LOCC)distinguishability of special maximally entangled states-generalized Bell state(GBS)is investigated.Firstly,the local distinguishability of generalized Bell states in arbitrary dimension is investigated.For this,we study the decomposition of a basis which consists of d2 number of generalized Pauli matrices.We discover that this basis is equal to the union of D number of different sets,where D=2/φ(d)(?)φ(i)+1 and φ is Eulerφ-function.Then we define the generator of the matrices in this decomposition,and exhibit an algorithm to calculate generators of a given set of matrices.This algorithm shows that generators of a given set can be calculated simply and efficiently.Thus,we show that a set L of GBSs can be distinguished by one-way LOCC if the cardinality of gL is less than Dφ(d),where gL is a set of generators of all the elements in difference set of a set L of GBSs.Secondly,in view of the generalized Bell state sets that the above methods can not judge its distinguishability,we show two other local distinguishable methods(only the composite dimension is discussed here):(1)A set of GBSs can be distinguished by oneway LOCC if its difference set is contained in Tq.(2)A set of GBSs can be distinguished by one-way LOCC if the intersection of its difference set and Mq is empty.Where Tq and Mq are only related to q,with q is a nontrivial factor of d.Then,we made a comparison with Takaaki el al.’s and Yuan el al.’s results,and found that our three methods can completely cover their results.Finally,we discuss the local unitary equivalence classification of generalized Bell state set proposed by predecessors,and found all the local distinguishability classification can be distinguished by us,then get the necessary and sufficient conditions of one-way local distinguishability of 3 generalized Bell state in arbitrary dimension,any generalized Bell states in C4(?)C4 and C5(?)C5.