Research Of Local Distinguishing The Orthogonal Product States With Entangled State As A Resource

Research on the relationship between quantum nonlocality and entanglement has become a hot issue of quantum information theory in recent years.In 1999,Bennett et al.first found nine orthogonal product states in C3(?)C3,which cannot be distinguished perfectly by local operations and classical communication.This phenomenon of nonlo-cality without entanglement illustrates that entanglement is not necessary for quantum nonlocality.However,it is well known that quantum entanglement which can serve as a resource to help distinguish quantum states is useful in quantum information processing.In 2008,Cohen locally distinguished certain classes of unextendible product bases(UPB)in Cm(?)Cn by using entanglement as a resource.He also at the same time left an open problem:What entanglement resources are sufficient and/or necessary for distinguishing quantum states with LOCC?In this thesis,we mainly study distinguishing orthogonal product states with LOCC by means of an entanglement state as a resource.Our main results are as follows:(i)Locally distinguishing orthogonal product states in bipartite quantum system with the least entanglement resource.(a)Firstly,we construct 2n-1 locally indistinguishable orthogonal product states in Cn(?)C4(n>4)and Cn(?)C5(n?5)respectively.Moreover,we construct a set of locally indistinguishable orthogonal product states with 2(n-1+2l)-8 members in Cn(?)C2l(n?2l>4)and a class of locally indistinguishable orthogonal product states with 2(n+2k+1)-7 members in Cn(?)C2k+1(n?2k+1>5)respectively.(b)Then,we exhibit a simple example,which explain the Cohen's method about entanglement as resources to distinguish the orthogonal product states by LOCC.(c)Along this way,we show that these classes of quantum states are distinguishable by LOCC only using a C2(?)C2 maximally entangled state.(ii)Distinguishing orthogonal product states by LOCC with an optimal entanglement resource state.We apply the definition of optimal resource state given by Bandyopadhyay,and then construct two sets of locally indistinguishable orthogonal product states in system C5(?)C5 and C7(?)C7.The maximally entangled state in system C3(?)C3 and C4(?)C4 are optimal resource states to distinguish the two sets of states by LOCC respectively.Finally,We study the maximum entangled state in the general bipartite system as a optimal resource to distinguish the orthogonal product states.(iii)Distinguishing multipartite orthogonal product states by LOCC with entanglement as a resource.We consider the set of LOCC indistinguishable multipar-tite product states constructed by Wang and show that they can be indeed distinguished by LOCC with entanglement as a resource.By separating the problem into the cases of even and odd partite states,we find out that each case is solved by considering the lower rank cases.(1)For the even-partite case,a set of LOCC indistinguishable multipartite product states are constructed by using multiple sets of locally indistinguishable bipartite orthog-onal product states.Therefore,the key of the case is to study the local distinguishability of a class of orthogonal product states in the bipartite system.(2)We are left to consider the odd-partite case,this is dealt with by first establishing the result for tripartite systems.The general odd partite quantum systems have com-posed of a tripartite case and even-partite cases.Therefore,it is easy to prove the local distinguishability of a class of orthogonal product states in general odd partite quantum systems.