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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 <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo>&amp;#x2297;<\/mml:mo><mml:mi>d<\/mml:mi><mml:mo>&amp;#x2297;<\/mml:mo><mml:mi>d<\/mml:mi><\/mml:math> for any <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo>&amp;#x2265;<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:math>, and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>3<\/mml:mn><mml:mo>&amp;#x2297;<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>&amp;#x2297;<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:math> 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.<\/jats:p>","DOI":"10.22331\/q-2022-01-05-619","type":"journal-article","created":{"date-parts":[[2022,1,5]],"date-time":"2022-01-05T17:14:42Z","timestamp":1641402882000},"page":"619","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":28,"title":["Strongly nonlocal unextendible product bases do exist"],"prefix":"10.22331","volume":"6","author":[{"given":"Fei","family":"Shi","sequence":"first","affiliation":[{"name":"School of Cyber Security, University of Science and Technology of China, Hefei, 230026, People&apos;s Republic of China"}]},{"given":"Mao-Sheng","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Physics, Southern University of Science and Technology, Shenzhen 518055, People&apos;s Republic of China"},{"name":"Department of Physics, University of Science and Technology of China, Hefei 230026, People&apos;s Republic of China"}]},{"given":"Mengyao","family":"Hu","sequence":"additional","affiliation":[{"name":"LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing 100191, People&apos;s Republic of China"}]},{"given":"Lin","family":"Chen","sequence":"additional","affiliation":[{"name":"LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing 100191, People&apos;s Republic of China"},{"name":"International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, People&apos;s Republic of China"}]},{"given":"Man-Hong","family":"Yung","sequence":"additional","affiliation":[{"name":"Department of Physics, Southern University of Science and Technology, Shenzhen 518055, People&apos;s Republic of China"},{"name":"Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, People&apos;s Republic of China"}]},{"given":"Yan-Ling","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Techonology, Dongguan University of Technology, Dongguan, 523808, People&apos;s Republic of China"}]},{"given":"Xiande","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People&apos;s Republic of China"}]}],"member":"9598","published-online":{"date-parts":[[2022,1,5]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"B. 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