The construction and local distinguishability of multiqubit unextendible product bases

TL;DR

This paper constructs a 7-qubit unextendible product basis (UPB) using unextendible orthogonal matrices, demonstrating its local indistinguishability and classifying its structure through graph theory, solving an open problem in quantum information.

Contribution

It introduces a novel 7-qubit UPB of size 11, solves an open problem, and analyzes its local indistinguishability and graph-theoretic structure.

Findings

Constructed a 7-qubit UPB of size 11.

Proved the UPB is locally indistinguishable in bipartite systems.

Classified the UPB's structure using nonisomorphic graphs.

Abstract

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.