The Local Distinguishability And Unextendibility Of Orthogonal Quantum States

The local distinguishability of quantum states is one of the basic problems in quan-tum information theory.It is helpful to explore the relationship between entanglement and nonlocality,and design quantum cryptographic protocols and so on.The local dis-tinguishing of the quantum states is that the participants randomly pick out a quantum state from some set of states which is known for all the participants,the participants can only determine the identity of the selected quantum state through local quantum opera-tion and classical communication.And because the unextendible product bases and the currently known unextendible maximally entangled bases are locally indistinguishable.In this paper,we mainly study the distinguishability and unextendibility of product states and maximally entangled states.The main research results are:(i)The local indistinguishable product states in multipartite quantum system.(a)First we introduce the graph representation of some quantum state,and give a set with 2n-1 local indistinguishable product states in Cm(?)Cn(4? m ? n)which are more concise than before.Second we use three-dimensional cubes to help us to construct the product states that can not be locally distinguished in tripartite quantum system.Finally the local indistinguishable product states in general mutipartite quantum system are given by using the results of bipartite and tripartite.(b)We study how to use entangled state as a resource to distinguish the local indistinguishable product states.This implies that entanglement could use to improve the power of the local distinguishability.(c)We show the relationship between the local distinguishable protocol and some special protocol tree.We prove that it is useless to improve the power of local distinguishability when increasing the number of the system or increasing the dimension of the system.Moreover,we show that the minimum number of local indistinguishable product states in multipartite quantum system is 4.(ii)The local distinguishability of the general Bell states.We extend some equations found by Fan in the prime dimension to arbitrary dimension.Our idea is based on the fact that the local distinguishability of the quantum states remain unchanged under local unitary transformation.Hence we can simplify the states by the local unitary operator before we start to distinguish them.Through these general equations and give some specific strategies analysis for special cases,we proved that any three general Bell states in Cd(?)Cd(d?4)are distinguishable.It is further confirmed the conjecture that any three maximally entangled states are locally distinguishable.(iii)For the unextendibility of the orthogonal quantum states.(a)The construction of the unextendible product bases.we give a method of construction the unextendible product basis in some large system by using the unextendible product basis in the subsystem.A series of unextendible product bases with different number of ele-ments in the bipartite quantum system are constructed by using this method.And we use the results of the bipartite quantum system to construct all kinds of unextendible product bases in multipartite quantum system.The unextendible product bases with vary number of elements can be used to construct some bounded entangled states with different rank which attracted lot of people's attention.On the other hand,the unextendible product basis is local indistinguishable which give a series of local indistinguishable product states.(b)The construction of the unextendible maximally entangled bases.First,we build up a one to one correspondence between the special unextendible maximally entangled bases in Cd(?)Cd and partial Hadamard matrices.Using this correspondence we first settle a conjecture about the Hadamard matrix:any(d-1)×d partial Hadamard matrix can be extended to a Hadamard matrix.Then we give an effective method to construct partial Hadamard matrix.We obtain that for any d there is a unextendible maximally entangled basis in Cd(?)Cd except for d = p or 2p,where p?3 mod 4 and p is a prime.We should notice that:all the known set of states about UMEB are local indistinguishable.