The Study Of Some Problems In Quantum Computation

Entanglement is one of the most basic concept in quantum computation and quantum information theory. Meanwhile,it is the key quantum source in quantum teleportation, quantum error correction and quantum secret sharing. The research for the high di-mensional entanglement is meaningful for the development of quantum information and application. This Thesis is mainly concerned with some basic feature of the maximally entangled states:the unextendibility and the indistinguishability.1. Unextendible maximally entangled bases(UMEB). Since the concept of maximally entangled states is dfined In Cd(?)Cd’(d<d’), we give an explicit construction of the UMEB and state the possibility that there might be more than two sets of UMEM with different number. And this give an answer to the question asked by the authors of [16]. For the case of d= d’, we start with an example and construct a set of 30 maximally entangled states in C60 C6 which is an UMEB. Then we generalize the method to higher dimensions. We obtain the following proposition:If there is an UMEB with N members in Cd (?) Cd, then we can construct an N members UMEB in Cqd (?) Cqd(whereN=(qd)2-(d2-N)). Finally, we study the equivalence of the UMEB we constructed.2.The LOCC indistinguishability of the maximally entangled states.First, we trans-fer the problem of LOCC distinguishability of quantum states into a semmidefinite pro-gramm. Then we find the feasible solutions to the programm. This feasible solution gives an upper bound of the successful possibility. Finally, we give d maximally entan-gled states which are LOCC indistinguishable in Cd(?)Cd. In addition, we also consider smaller sets of LOCC indistinguishable maximally entangled states. We obtain that for any d=k2t-2, if p is the least prime number divides by k, then when p< 2t-3,there always exists L(L<d) maximally entangled states can not be LOCC distinguished.