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arXiv:2411.18036 · Journal of Mathematical Physics

Unextendible and strongly uncompletable product bases

Xiao-Fan Zhen, Hui-Juan Zuo, Fei Shi, Shao-Ming Fei

Published Nov 1, 2024• 2 citations
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Abstract

In 2003, DiVincenzo et al. put forward the question of whether there exists an unextendible product basis (UPB) that is an uncompletable product basis in every bipartition [DiVincenzo et al. Commun. Math. Phys. 238, 379–410 (2003)]. Recently, Shi et al. presented a UPB in tripartite systems that is also a strongly uncompletable product basis (SUCPB) in every bipartition [Shi et al. New J. Phys. 24, 113025 (2022)]. However, whether there exist UPBs that are SUCPBs in only one or two bipartitions remains unknown. We provide a sufficient condition for the existence of SUCPBs based on a quasi U-tile structure. We analyze all possible cases about the relationship between UPBs and SUCPBs in tripartite systems. In particular, we construct a UPB with smaller size d3 − 3d2 + 1 in Cd⊗Cd⊗Cd, which is an SUCPB in every bipartition and has a smaller cardinality than the existing one.

DOI
10.1063/5.0195550

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In 2003, DiVincenzo et al. put forward the question of whether there exists an unextendible product basis (UPB) that is an uncompletable product basis in every bipartition [DiVincenzo et al. Commun. Math. Phys. 238, 379–410 (2003)]. Recently, Shi et al. presented a UPB in tripartite systems that is also a strongly uncompletable product basis (SUCPB) in every bipartition [Shi et al. New J. Phys. 24, 113025 (2022)]. However, whether there exist UPBs that are SUCPBs in only one or two bipartitions remains unknown. We provide a sufficient condition for the existence of SUCPBs based on a quasi U-tile structure. We analyze all possible cases about the relationship between UPBs and SUCPBs in tripartite systems. In particular, we construct a UPB with smaller size d3 − 3d2 + 1 in Cd⊗Cd⊗Cd, which is an SUCPB in every bipartition and has a smaller cardinality than the existing one.
Abstract Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging from quantum nonlocality to quantum cryptography. A generalized concept called uncompletable product basis also attracts much attention. In this paper, we find some unextendible product bases that are uncompletable product bases in every bipartition, which answers a 19 year-old open question proposed by DiVincenzo et al (2003 Commun. Math. Phys. 238 379–410). As a consequence, we connect such unextendible product bases to local hiding of information, positive-partial-transpose entangled states and genuinely entangled states. Furthermore, we give a sufficient condition for the existence of an unextendible product basis that is still unextendible in every bipartition, and the existence of such a UPB is another open question proposed by Demianowic et al (2018 Phys. Rev. A 98 012313). Our results advance the understanding of the geometry of unextendible product bases.
Abstract Unextendible product basis (UPB) is one of the most fundamental concepts in the field of quantum information theory, and its constructions attract much attention. Recently, Shi et al (2020 Phy. Rev. A 101 062329) constructed UPBs via a tile structure approach. They showed that a tile structure in an m × n grid with certain properties leads to explicit constructions of UPBs in C m ⊗ C n . The size of the UPB is m n − s + 1 if the tile structure contains exactly s tiles. In this paper, we deeply analyze the extent of the tile structure approach, by showing that a desired tile structure in an m × m grid contains at most ⌊ m 2 4 ⌋ + m tiles, and thus the smallest size of a UPB in C m ⊗ C m constructed by this approach is m 2 − ⌊ m 2 4 ⌋ − m + 1 . Such tile structures and UPBs are referred to as extremal tile structures and extremal UPBs. We provide algorithms both for constructing possible candidates of extremal tile structures and checking their validity. For applications, we connect the extremal UPBs to local irreducibility and bound entangled states with large ranks.
Unextendible product bases (UPBs) are interesting members of a family of orthogonal product bases. Here, we investigate the construction of 3-qudit UPBs with strong nonlocality. First, a UPB set in C 3 ⊗ C 3 ⊗ C 3 of size 19 is presented based on the shift UPBs. By mapping the system to a Rubik’s cube, we provide a general method of constructing UPBs in Cd ⊗ Cd ⊗ Cd of size ( d – 1 )3 + 2d + 5, whose corresponding Rubik’s cube is composed of four parts. Second, for the more general case where the dimensions of parties are different, we extend the classical tile structure to the 3-qudit system and propose the tri-tile structure. By means of this structure, a C 4 ⊗ C 4 ⊗ C 5 system of size 38 is obtained based on a C 3 ⊗ C 3 ⊗ C 4 system of size 19. Then, we generalize this approach to the C d 1 ⊗ C d 2 ⊗ C d 3 system which also consists of four parts. Our research provides a positive answer to the open question raised in by Halder et al. [Phys. Rev. Lett. 122 040403 (2019)], indicating that there do exist UPBs that can exhibit strong quantum nonlocality without entanglement.
Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. A famous open question is whether there exist genuinely unextendible product bases (GUPBs), namely multipartite product bases that are unextendible with respect to every possible bipartition. Here we shed light on this question by providing a characterization of UPBs and GUPBs in terms of orthogonality graphs. Building on this connection, we develop a method for constructing UPBs in low dimensions, and we derive a lower bound on the size of any GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs. Finally, we discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.
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.
Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. Here we provide an equivalent characterization of UPBs in graph-theoretic terms. Different from previous graph-theoretic investigations of UPBs, which focused mostly on the orthogonality relations between different product states, our characterization includes a graph-theoretic reformulation of the unextendibility condition. Building on this characterization, we develop a constructive method for building UPBs in low dimensions and shed light on the open question of whether there exist genuinely unextendible product bases (GUPBs), that is, multipartite product bases that are unextendible with respect to every possible bipartition. We derive a lower bound on the size of any candidate GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs and discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension. Finally, we apply our characterization to the problem of distinguishing UPB states under local operations and classical communication, deriving a necessary condition for reliable discrimination in the asymptotic limit.