The Research On Some Key Problems In Local Distinguishability Of Quantum States

The problem about local distinguishability of quantum states is one of basic research contents in quantum information theory. It has very important significance both in theory and in practical application. In theory, it is an effective way to research the relationship between quantum entanglement and nonlocality. In practical application, one can use cheap LOCC to reduce quantum communication and global operations, so it can save costs. In this thesis, we focus on some open but unsolved questions about local distinguishability of quantum states.Details are as follows.(1) About local distinguishability of orthogonal quantum states in 3-qubit system, we first solve the question proposed by Duan, what the maximal number of locally distinguishable states in the locally indistinguishable subspaces is. Then we show a necessary condition for local distinguishability of three-qubit quantum states. According the necessary condition, we find that a set containing more than 8-N three-qubit tripartite entangled states is locally indistinguishable, in other words, a set containing fewer than 2V-8 orthogonal product states is locally indistinguishable, where N is the number of quantum states in the set. Finally, applying the characterization of orthogonal product states, we obtain two methods to discriminate mixed states with some constraints.(2) About characterizing unextendible product bases in qutrit-ququad system, we study the UPB in qutrit-ququad system and find that there only exist six, seven and eight-state UPB. We completely characterize the six-state and seven-state UPB. For eight-state UPB, seven classes of UPB are found. As auxiliary results, we study the distinguishability of qutrit-ququad UPB by separable measurements, and find that there exists a UPB that cannot be distinguished.(3) About local unambiguous discrimination between multipartite quantum states, We demonstrate that any no more than max{di} linearly independent multipartite pure quantum states can be locally unambiguously discriminated, where the space spanned by the set can be expressed in the irreducible form (?)i=1N, di and di is dimension of the ith party. That is, max{di} is an upper bound. We also show that the bound is tight, namely there exists a set of max{di｝＋1 states, in which at least one of the states cannot be locally unambiguously discriminated. Our result gives the reason why n-qubit system is the only exception when any three quantum states are locally unambiguously distinguished.(4) About application of local distinguishability theory in quantum secret sharing, we investigate the distinguishability of orthogonal multipartite entangled states in d-qudit system by restricted local operations and classical communication. According to these properties,we propose a standard (2,n)-threshold quantum secret sharing scheme(called LOCC-QSS scheme), which solves the open question proposed by Rahaman. On the other hand, we find that all the existing (k, n)-threshold LOCC-QSS schemes are imperfect (or "ramp"), i.e., unauthorized groups can obtain some information about the shared secret. Furthermore, we present a (3,4)-threshold LOCC-QSS scheme which is close to perfect.