The construction and weakly local indistinguishability of multiqubit unextendible product bases

An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we first construct a 7-qubit UPB of size 11 and prove its existence. It turns out that the UPB corresponds to a complete graph with 11 vertices constructed by three sorts of nonisomorphic graphs. Taking the graphs as product vectors, we show that they are in three different orbits up to local unitary equivalence. Next, we also provide more methods to construct UPBs and present the number of sorts of nonisomorphic graphs of complete graphs corresponding to some known UPBs and their orbits. Moreover, we graph-theoretically show that the UPB is weakly locally indistinguishable in the bipartite systems of two (three) qubits and five (four) qubits, respectively. This result is applied in secret sharing.