Extension of the unextendible product bases of 5-qubit under coarsening the system

It has been shown that there exist unextendible product bases (UPBs) with sizes of 6, 8, 9, and 10 in the 5-qubit system. These product vectors are not UPB any more when they are regarded as product vectors in the coarsened systems (i.e., some subsystems are merged, e.g., if two subsystems are merged, the system becomes 2 circle times 2 circle times 2 circle times 4). In this paper, we show that these product states can be expanded to UPBs in the coarsened system by adding some product vectors. Comparing with the existing UPBs, we get a series of novel UPBs which have different sizes or have the same size as that of the existing ones but are not equivalent to them.