Orthogonal product bases of four qubits

TL;DR

This paper classifies orthogonal product bases of four-qubit systems by reducing the problem to a combinatorial one and identifying 33 families of such bases, providing a comprehensive understanding of their structure.

Contribution

It reduces the classification of four-qubit OPBs to a combinatorial problem and explicitly solves it, identifying 33 families of bases.

Findings

33 multiparameter families of OPBs identified

Each four-qubit OPB is equivalent to one in these families

Classification simplifies understanding of four-qubit bases

Abstract

An orthogonal product basis (OPB) of a finite-dimensional Hilbert space $H=H_1\otimes H_2\otimes\cdots\otimes H_n$ is an orthonormal basis of $H$ consisting of product vectors $x_1\otimes x_2\otimes\cdots\otimes x_n$. We show that the problem of classifying 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.