The Construction Of Unextendible Entangled Bases With Schmidt Number And Mutually Unbiased Bases

Quantum information theory is an interdisciplinary subject of quantum mechan-ics and information science.It is an emerging frontier science that studies information processing based on the basic principles of quantum mechanics.We use quantum states to represent the physical state of quantum system in the field of quantum information.The processing and reading process of quantum information is inseparable from the quantum state.Generally speaking,the quantum state can be divided into one single quantum system and multipartite quantum system according to the number of par-ticles.Multipartite quantum systems can produce a peculiar phenomenon based on the principle of superposition of quantum states,namely entanglement.As the most significant feature of quantum mechanics differs from classical mechanics,quantum en-tanglement has become a hot issue in the field of quantum information.Therefore,in order to describe entanglement mathematically,the study of various bases structures in the state space are particularly important.To determine an unknown quantum state,it is necessary to perform multiple quantum measurements.The mutually unbiased basis is just the best set of measurement basis that minimizes the reconstruction of the quantum state.This shows that the mutually unbiased basis plays an important role in quantum information.The paper mainly discusses the structure of the unextendible entangled bases with fixed Schmidt number k(UEBk)and the mutually unbiased entangled bases with fixed Schmidt number k(MUSEBks)in bipartite system.We mainly study the above content in three aspects,we first show the general construction of the UEBk with fixed members,then we provide the construction of the UEBk with different members,finally,present the mutually unbiased unextendible entangled bases with Schmidt number k(MUUEBks).The specific content of the paper is arranged as follows:First,we introduce the background and history of the UEBk and MUB,some necessary basic knowledge and concepts of quantum information theory related to the paper are introduced.Then,we use the graph to study the general construction of the UEBk with fixed members in bipartite system.The construction of the UEBk are discussed in two cases by the Schmidt number k is the multiple of the space dimension d and or not.After ignoring the elements in the whole row and column,we present the method of unequal division of the residual space dimensions,the general expression and necessary proofs of the UEBk are given,the construction of UEBks in C3(?)C7 and(C4(?)(C10 are also shown.We study the construction of different members UEBks in bipartite system which members are bigger than the above cases in a certain interval.We construct the UEBk by ignoring the scattered elements and the generalized weighting matrices.We discuss these ignored elements separately by the way of occupying different rows and columns for normalize.Different ways of rotating bases are also be given to solve it in each cases.The general expression and necessary proofs of the UEBk are given,the construction of UEBks in(C5(?)C6 and C6(?)C6 are also shown.We study mutually unbiased SEBks in bipartite system.We discuss the unbiased-ness between the SEB3s in C4(?)C4,then we find the form satisfied by the transition matrix which satisfies the unbiased condition.Finally,we summarize the paper.