The Structure Of Product Bases Of Several Kinds Of Spaces

The unbiased basis is first proposed to solve the problem of reconstruction of quan-tum states.The reason is that there are lots of overlaps between the measurement results in the process of quantum reconstruction,i.e.there may be large redundancy in these results.Because the reconstruction process of states based on mutually unbiased bases is free of information waste,we refer to this scheme as the optimal scheme.The known mutually unbiased bases are the form of product bases in dimension 6,so the structure of product bases had been widely concerned.Based on this reason,this paper mainly studies the structure of product bases.1.We study the structure of product bases of 2(?)n using the partition of integer.we obtain all the product bases of a bipartite system 2(?)n.It popularized the structure of all the product bases of 2(?)2 and 2(?)3.In addition,we research the product bases of m(?)n,and obtain the structure of all the product bases of 3(?)3.It gives an answer to the conjecture in case of d = 2n and d = 3× 3.2.The product bases of 2(?)2 and 2(?)3 are extended to 2(?)2(?)2 and 2(?)2(?)3,and we can obtain the structure of product bases of 2(?)2(?)2 and 2(?)2(?)3.Finally,we think this method can be extended to 2(?)k(?)3(?)l and also have similar results.3.We simplify the distinguishability of product bases of 2(?)n using LOCC.