Strongly nonlocal unextendible product bases do exist

TL;DR

This paper proves the existence of strongly nonlocal unextendible product bases in tripartite systems of dimension three or higher, demonstrating phenomena of strong quantum nonlocality without entanglement and solving open questions in the field.

Contribution

The authors develop new techniques to show that certain UPBs are locally irreducible in every bipartition, establishing their strong nonlocality without entanglement in tripartite systems.

Findings

Existence of UPBs that are locally irreducible in every bipartition for all d≥3

Minimum dimension for such UPBs is 3×3×3

These UPBs exhibit strong quantum nonlocality without entanglement

Abstract

A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving local measurement can be performed to this set. In general, it is difficult to show that such an orthogonality-preserving local measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully show the existence of unextendible product bases (UPBs) that are locally irreducible in every bipartition in $d\otimes d\otimes d$ for any $d\geq 3$, and $3\otimes3\otimes 3$ achieves the minimum dimension for the existence of such UPBs. These UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Our result solves an open question given by Halder \emph{et al.} [Phys. Rev. Lett. \textbf{122}, 040403 (2019)] and Yuan \emph{et al.} [Phys. Rev. A \textbf{102}, 042228 (2020)]. It also sheds new light on the connections between UPBs and strong quantum nonlocality.