Constructions Of Mutually Unbiased Unextendible Entangled Bases

Quantum entanglement plays a very important role in quantum computation and quantum information processing,but its properties and mathematical structure have not been fully described.Unbiasedness is an important method to describe quantum entanglement.In recent years,many researches have been carried out on product basis,unextendible product basis,maximally entangled basis and unextendible maximally entangled basis,and the structures of all kinds of mutually unbiased bases have been explored step by step.The paper mainly studies the constructions of mutually unbiased unextendible maximally entangled bases in bipartite quantum systems and mutually unbiased unextendible genuinely multiparty entangled bases in multipartite quantum systems.The results mainly include the following three aspects:(1)we present two sets of mutually unbiased unextendible maximally entangled bases for bipartite quantum system C^d(?)C^d'(d<d')with prime number d'.(2)The constructions of unextendible genuinely entangled basis for tripartite quantum system C^d1(?)C^d2(?)C^d3(d₁?d₂<d₃)is provided.Two sets of unextendible genuinely entangled bases are constructed by Fourier matrix and extended into orthonormal bases by adding fully separable states.Then,it is proved that the two bases are mutually unbiased.(3)we also generalize the construction method from tripartite quantum systems to N-partite quantum systems.Based on the properties of the Fourier matrix,we construct mutually unbiased unextendible genuinely multiparty entangled bases in multipartite system(?)_k=1^N C^dk(d₁?…?d_N-1<d_N).In order to explain each construction,the corresponding examples and strict proofs are proposed.