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⊗d⊗d for any d≥3, and 3⊗3⊗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 et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. It also sheds new light on the connections between UPBs and strong quantum nonlocality.
Highlights
Quantum state discrimination has attracted more and more attention in recent years
In Propositions 1 and 2, we successfully show the existence of strongly nonlocal unextendible product bases (UPBs) in 3⊗3⊗ 3 and 4⊗4⊗4, respectively
Based on the two strongly nonlocal UPBs, in Theorem 1, we show that strongly nonlocal UPBs do exist for any tripartite system d ⊗ d ⊗ d (d ≥ 3)
Summary
Quantum state discrimination has attracted more and more attention in recent years. Consider a composite quantum system prepared in a state from a known. A set of orthogonal states is strongly nonlocal if it is locally irreducible in every bipartition They showed such a phenomenon by presenting two strongly nonlocal orthogonal product bases in 3⊗3⊗3 and 4 ⊗ 4 ⊗ 4, respectively. Many efforts have been made in the strong quantum nonlocality, the existence of strongly nonlocal UPBs remains unknown This is an open question in Refs. We develop two useful techniques in Lemmas 1 and 2 to prove that a set of orthogonal product states is strongly nonlocal. By utilizing such techniques, in Propositions 1 and 2, we successfully show the existence of strongly nonlocal UPBs in 3⊗3⊗ 3 and 4⊗4⊗4, respectively. Based on the two strongly nonlocal UPBs, in Theorem 1, we show that strongly nonlocal UPBs do exist for any tripartite system d ⊗ d ⊗ d (d ≥ 3)
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