Constructions Of Mutually Unbiased Unextendible Maximally Entangled Bases

In quantum information processing,quantum state is the concrete manifestation of quantum information.In order to obtain the information of the unknown quantum state,it is necessary to measure the quantum state.Mutually unbiased bases provide a optimal choice for measurement,which has attracted extensive attention of scholars,and its composition is becoming more and more diversified,including product bases,maximally entangled bases,special entangled basis with schmidt number k,unextendible product basis and unextendible maximally entangled bases,and so on.Based on the unextendible maximally entangled bases,this paper mainly study the construction of mutually unbiased basis in bipartite systems Cd(?)Cd’(d<d’),and the main results are as follows:(1)When d’ is an odd prime number,two sets of mutually unbiased unextendible maximally entangled bases are constructed by using unitary operators over finite fields Fp.(2)When d’=4k,where k is a positive integer,we dealt with the problem in three cases:(ⅰ)d’=p+1,where p is a prime number,based on the Paley construction of binary sequences,two sets of mutually unbiased unextendible maximally entangled bases are presented;(ⅱ)d’=q+1,where q=p(p+2)and p,p+2 are prime numbers,based on binary sequences constructed by twin-prime,two sets of mutually unbiased unextendible maximally entangled bases are constructed;(ⅲ)d’=2m,where m≥2 is a positive integer,using the Singer construction of binary sequences with ideal autocorrelation,two sets of mutually unbiased unextendible maximally entangled bases are obtained.(3)When d’=q,where q is an odd prime power,two sets of mutually unbiased unextendible maximally entangled bases are constructed by a generalized m sequence with ideal autocorrelation.In order to better explain each of our constructions,the corresponding concrete examples are shown in each part.