Orthogonal product bases of four qubits

Abstract

An orthogonal product basis (OPB) of a finite-dimensional Hilbert space is an orthonormal basis of consisting of product vectors . We show that the problem of constructing the OPBs of an n-qubit system can be reduced to a purely combinatorial problem. We solve this combinatorial problem in the case of four qubits and obtain 33 multiparameter families of OPBs. Each OPB of four qubits is equivalent, under local unitary operations and qubit permutations, to an OPB belonging to at least one of these families.