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Strong nonlocal sets of UPB (2106.08699v2)

Published 16 Jun 2021 in quant-ph

Abstract: The unextendible product bases (UPBs) are interesting members from the family of orthogonal product states. In this paper, we investigate the construction of 3-qubit UPB with strong nonlocality of different sizes. First, a UPB set in ${{C}{3}}\otimes {{C}{3}}\otimes {{C}{3}}$ of size 12 is presented based on the Shifts UPB, the structure of which is described by mapping the system to a $3\times 3\times 3$ Rubik's Cube. After observing the orthogonal graph of each qubit, we provide a general method of constructing UPB in ${{C}{d}}\otimes {{C}{d}}\otimes {{C}{d}}$ of size ${{\left( d-1 \right)}{3}}+3\left( d-2 \right)+1$. Second, for the more general case where the dimensions of qubits are different, we extend the tile structure to 3-qubit system and propose a Tri-tile structure for 3-qubit UPB. Then, by means of this structure, a ${{C}{4}}\otimes {{C}{4}}\otimes {{C}{5}}$ system of size 30 is obtained based on a ${{C}{3}}\otimes {{C}{3}}\otimes {{C}{4}}$ system. Similarly, we generalize this approach to ${{C}{{{d}_{1}}}}\otimes {{C}{{{d}_{2}}}}\otimes {{C}{{{d}_{3}}}}$ system which has a similar composition to ${{C}{d}}\otimes {{C}{d}}\otimes {{C}{d}}$. Our research provides a positive answer to the open questions raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist multi-qubit UPBs that can exhibit strong quantum nonlocality without entanglement.

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