Constructing the three-qudit unextendible product bases with strong nonlocality

The unextendible product bases (UPB) are interesting members of the family of orthogonal product bases. In this paper, we investigate the construction of 3-qudit UPB with strong nonlocality. First, a UPB set in ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{3}}$ of size 19 is presented based on the Shifts UPB. By mapping the system to a Rubik's Cube, we provide a general method of constructing UPB in ${{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}$ of size ${{\left(d-1 \right)}^{3}}+2d+5$, 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 ${{C}^{4}}\otimes {{C}^{4}}\otimes {{C}^{5}}$ system of size 38 is obtained based on a ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{4}}$ system of size 19. Then, we generalize this approach to ${{C}^{{{d}_{1}}}}\otimes {{C}^{{{d}_{2}}}}\otimes {{C}^{{{d}_{3}}}}$ system which also consists of four parts. Our research provides a positive answer to the open question raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist UPB that can exhibit strong quantum nonlocality without entanglement.

Strongly nonlocal unextendible product bases do exist

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.

Strong quantum nonlocality for unextendible product bases in heterogeneous systems

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 dAXdBxdC and dAxdBxdCxdD for dA, dB, dC, dD ≧3, and we also show that these UPBs have strong quantum nonlocality, which answers an open question given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)] for any possible three and four-partite systems. Furthermore, we propose an entanglement-assisted protocol to locally discriminate the UPB in 3x3x4, and it consumes less entanglement resource than the teleportation-based protocol. Our results build the connection between strong quantum nonlocality and UPBs.

A Quantum Dual-Signature Protocol Based on SNOP States without Trusted Participant

Quantum dual-signature means that two signed quantum messages are combined and expected to be sent to two different recipients. A quantum signature requires the cooperation of two verifiers to complete the whole verification process. As an important quantum signature aspect, the trusted third party is introduced to the current protocols, which affects the practicability of the quantum signature protocols. In this paper, we propose a quantum dual-signature protocol without arbitrator and entanglement for the first time. In the proposed protocol, two independent verifiers are introduced, here they may be dishonest but not collaborate. Furthermore, strongly nonlocal orthogonal product states are used to preserve the protocol security, i.e., no one can deny or forge a valid signature, even though some of them conspired. Compared with existing quantum signature protocols, this protocol does not require a trusted third party and entanglement resources.

Novel quantum group signature scheme based on orthogonal product states

In this paper, we present a quantum group signature (QGS) scheme based on orthogonal product states (OPSs) that cannot be perfectly distinguished by local operations and classical communication. Our scheme has all the properties of QGS, including unforgeability, undeniability, traceability, verifiability and anonymity. These properties can guarantee the security of the scheme. More importantly, different particles of a product state that comes from a nonlocal set are transmitted separately, thus the information that is encoded in the product state will not be leaked. Security and efficiency analysis of the scheme show that our scheme is secure and efficient.