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arXiv:2106.08699

Strong nonlocal sets of UPB

Bichen Che, Zhao Dou, Min Lei, Yixian Yang

Published Jun 16, 2021• 2 citations
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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.

DOI
10.48550/arxiv.2106.08699

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Abstract A set of multipartite orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. It is known that unextendible product bases (UPBs) can show the phenomenon of quantum nonlocality without entanglement. Thus it is interesting to investigate the strong quantum nonlocality for UPBs. Most of the UPBs with the minimum size cannot demonstrate strong quantum nonlocality. In this paper, we construct a series of UPBs with different large sizes in d A ⊗ d B ⊗ d C and d A ⊗ d B ⊗ d C ⊗ d D for d A , d B , d C , d D ⩾ 3, and we also show that these UPBs have strong quantum nonlocality, which answers an open question given by Halder et al (2019 Phys. Rev. Lett. 122 040403) and Yuan et al (2020 Phys. Rev. A 102 042228) for any possible three and four-partite systems. Furthermore, we propose an entanglement-assisted protocol to locally discriminate the UPB in 3 ⊗ 3 ⊗ 4, and it consumes less entanglement resource than the teleportation-based protocol. Our results build the connection between strong quantum nonlocality and UPBs.
We completely characterize the condition when a tile structure provides an unextendible product basis (UPB), and construct UPBs of different large sizes in Cm⊗Cn\mathbb{C}^m\otimes\mathbb{C}^nCm⊗Cn for any n≥m≥3n\geq m\geq 3n≥m≥3. This solves an open problem in [S. Halder et al., Phys. Rev. A 99, 062329 (2019)]. As an application, we show that our UPBs of size (mn−4⌊m−12⌋)(mn-4\lfloor\frac{m-1}{2}\rfloor)(mn−4⌊2m−1​⌋) in Cm⊗Cn\mathbb{C}^m\otimes\mathbb{C}^nCm⊗Cn can be perfectly distinguished by local operations and classical communications assisted with a ⌈m2⌉⊗⌈m2⌉\lceil\frac{m}{2}\rceil\otimes\lceil\frac{m}{2}\rceil⌈2m​⌉⊗⌈2m​⌉ maximally entangled state.
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 measurement can be performed to this set. In general, it is difficult to show such a orthogonality-preserving measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully construct orthogonal product sets (OPSs) in any possible three, four and five-partite systems that are locally irreducible in every bipartition. Moreover, we also show the existence of unextendible product bases (UPBs) with such properties in d⊗d⊗dd\otimes d\otimes dd⊗d⊗d for any d≥3d\geq 3d≥3. Therefore, these OPSs and UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Both results solve the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. Our results shed a new light on the strong quantum nonlocality without entanglement.
Unextendible product bases (UPBs) are important structures in quantum information theory, with applications to completely entangled subspaces, bound entanglement, and local indistinguishability. Since many properties and applications of UPBs are closely related to their cardinalities, one of the central problems in the study of UPBs is to determine whether UPBs of prescribed sizes exist in a given multipartite system. In this paper, we introduce a SAT-assisted framework based on decompositions of the NNN-dimensional hypercube. We define ONO_NON​-tile decompositions and prove a tile-to-UPB theorem: every ONO_NON​-tile decomposition induces a UPB through a construction based on tile-wise Fourier product bases and a global stopper state. We then encode the search for such decompositions as a Boolean satisfiability (SAT) problem and use SAT solvers to generate explicit instances. In terms of verification, we also implement a UPB verification algorithm based on local orthogonality graphs and unsaturated subspaces. The algorithm can be used to determine whether an arbitrary finite set of product states forms a UPB. Using this framework, we obtain UPBs of several sizes in some tripartite and quadripartite systems, including sizes 13,14,…,2313,14,\ldots,2313,14,…,23 in C3⊗C3⊗C3\mathbb C^3\otimes\mathbb C^3\otimes\mathbb C^3C3⊗C3⊗C3. Moreover, the small-dimensional instances obtained here can serve as seed UPBs for recursive constructions, leading to further examples in larger multipartite systems.
We completely characterize the condition when a tile structure provides an unextendible product basis (UPB) and construct UPBs of different large sizes in ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. In particular, we show that there exists a UPB of size $(mn\ensuremath{-}4\ensuremath{\lfloor}\frac{m\ensuremath{-}1}{2}\ensuremath{\rfloor})$ in ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ for any $n\ensuremath{\ge}m\ensuremath{\ge}3$, which solves an open problem [S. Halder et al., Phys. Rev. A 99, 062329 (2019)]. As an application, we show that this class of UPBs can be perfectly distinguished by local operations and classical communications assisted with a $\ensuremath{\lceil}\frac{m}{2}\ensuremath{\rceil}\ensuremath{\bigotimes}\ensuremath{\lceil}\frac{m}{2}\ensuremath{\rceil}$ maximally entangled state.