The construction and local distinguishability of multiqubit unextendible product bases

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 graphtheoretically 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.

Constructing the three-qudit unextendible product bases with strong nonlocality

“…Nevertheless, most of the current efforts are devoted to the construction of 2-qubit UPBs, [22][23][24] while a little progress has been made on multiqudit UPBs. [25,26] Chen et al [22] investigated the minimum size of UPBs with local dimension of 2, and analyzed the proposed sets using orthogonal graphs. Bej et al [23] proposed that a set of reducible unextendible product states (UPSs) with high dimension can be obtained by adding several orthogonal product states on the bases of a set of UPSs with low dimension in a bipartite system.…”

Strong nonlocal sets of UPB

“…[19][20][21] has shown that it can be used in the production of bound entangled (BE) states and some special bipartite entangled states that remain positive under partial transpose (PPT). Nevertheless, most of the current efforts are devoted to the construction of 2-qubit UPBs, while little progress has been made on multi-qubit UPBs [22][23][24][25][26]. Chen et al [22] investigated the minimum size of UPB with local dimension equals 2, and analyzed the proposed sets using orthogonal graphs.…”