  An important problem in quantum information is to construct multiqubit
unextendible product bases (UPBs). By using the unextendible orthogonal
matrices, we construct a 7-qubit UPB of size 11. It solves an open problem in
[Quantum Information Processing 19:185 (2020)]. Next, we graph-theoretically
show that the UPB is locally indistinguishable in the bipartite systems of two
qubits and five qubits, respectively. 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. Moreover, we also present the
number of sorts of nonisomorphic graphs of complete graphs of some known UPBs
and their orbits.
