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We studied the construction problem of the unextendible product basis (UPB).We mainly give a method to construct a UPB of a quantum system through the UPBsof its subsystem. Using this method and the UPBs which are known for us, weconstruct different kinds of UPBs in general bipartite quantum system. Then weuse these UPBs to construct a family of UPBs in multipartite quantum system.The UPBs can be used to construct the bound entangled states with differentranks.

Constructing unextendible product bases from the old ones