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.
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
C3⊗C3⊗C3 of size 12 is presented based on the Shifts UPB, the structure of which is described by mapping the system to a
3×3×3 Rubik's Cube. After observing the orthogonal graph of each qubit, we provide a general method of constructing UPB in
Cd⊗Cd⊗Cd of size
(d−1)3+3(d−2)+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
C4⊗C4⊗C5 system of size 30 is obtained based on a
C3⊗C3⊗C4 system. Similarly, we generalize this approach to
Cd1⊗Cd2⊗Cd3 system which has a similar composition to
Cd⊗Cd⊗Cd. 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.
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.
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.
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⊗d for any
d≥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.
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 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 d ⊗ d ⊗ d for any d ≥ 3 , and 3 ⊗ 3 ⊗ 3 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.
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 for any
n≥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⌊2m−1⌋) in
Cm⊗Cn can be perfectly distinguished by local operations and classical communications assisted with a
⌈2m⌉⊗⌈2m⌉ maximally entangled state.