# Strongly nonlocal unextendible product bases do exist

Fei Shi1, Mao-Sheng Li2,3, Mengyao Hu4, Lin Chen4,5, Man-Hong Yung2,6, Yan-Ling Wang7, and Xiande Zhang8

1School of Cyber Security, University of Science and Technology of China, Hefei, 230026, People's Republic of China
2Department of Physics, Southern University of Science and Technology, Shenzhen 518055, People's Republic of China
3Department of Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
4LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing 100191, People's Republic of China
5International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, People's Republic of China
6Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, People's Republic of China
7School of Computer Science and Techonology, Dongguan University of Technology, Dongguan, 523808, People's Republic of China
8School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People's Republic of China

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### 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 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.

The most well-known form of quantum nonlocality-Bell nonlocality, arises from entangled states. Bennett et al. [Phys. Rev. A 59, 1070 (1999)] showed that product states can exhibit nonlocal properties in a way which is fundamentally different from Bell nonlocality. In fact, they presented nine orthogonal product states of two qutrit systems such that these states are indistinguishable by using local operation and classical communication.

More recently, Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] introduced a strongest form of nonlocality and they presented two examples in tripartite systems to show the existence of such kind of strong nonlocality. Although there are several works on this kind of strong nonlocality, the current method is very difficult to verify the strong nonlocality because of a large number of calculations.

Our main contribution is to develop two basic techniques which are very useful for showing that a set of orthogonal product states is strongly nonlocal. By using these techniques, we successfully show the existence of strongly nonlocal unextendible product bases in $d\otimes d\otimes d$ for $d\geq 3$. This 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 strong quantum nonlocality without entanglement.

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### Cited by

[1] Fei Shi, Mao-Sheng Li, Mengyao Hu, Lin Chen, Man-Hong Yung, Yan-Ling Wang, and Xiande Zhang, "Strong quantum nonlocality from hypercubes", arXiv:2110.08461.

[2] Yan-Ling Wang, Mao-Sheng Li, and Man-Hong Yung, "Graph-connectivity-based strong quantum nonlocality with genuine entanglement", Physical Review A 104 1, 012424 (2021).

[3] Fei Shi, Mao-Sheng Li, Lin Chen, and Xiande Zhang, "Strong quantum nonlocality for unextendible product bases in heterogeneous systems", Journal of Physics A Mathematical General 55 1, 015305 (2022).

[4] Mao-Sheng Li and Zhu-Jun Zheng, "Genuine hidden nonlocality without entanglement: from the perspective of local discrimination", arXiv:2111.02891.

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