Strongly nonlocal unextendible product bases do exist

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 $d\ge 3$, and $3\otimes 3\otimes 3$ 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.

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• Genuine hidden nonlocality without entanglement: from the perspective of local discrimination
• Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$
• Small set of orthogonal product states with nonlocality
• Strong quantum nonlocality from hypercubes
• Negative result about the construction of genuinely entangled subspaces from unextendible product bases
• Information locking and its resource efficient extraction
• Constructing unextendible product bases from multiqubit ones
• Application of Ramsey theory to localization of set of product states via multicopies