Strong nonlocal sets of UPB

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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Cited by in corpus (1)

• Strong quantum nonlocality without entanglement in every $(n-1)$-partition