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Paper 1
The 27-qubit Counterexample to the LU-LC Conjecture is Minimal
Nathan Claudet
- Year
- 2026
- Journal
- arXiv preprint
- DOI
- arXiv:2603.25219
- arXiv
- 2603.25219
It was once conjectured that two graph states are local unitary (LU) equivalent if and only if they are local Clifford (LC) equivalent. This so-called LU-LC conjecture was disproved in 2007, as a pair of 27-qubit graph states that are LU-equivalent, but not LC-equivalent, was discovered. We prove that this counterexample to the LU-LC conjecture is minimal. In other words, for graph states on up to 26 qubits, the notions of LU-equivalence and LC-equivalence coincide. This result is obtained by studying the structure of 2-local complementation, a special case of the recently introduced r-local complementation, and a generalization of the well-known local complementation. We make use of a connection with triorthogonal codes and Reed-Muller codes.
Open paperPaper 2
Strongest quantum nonlocality in $N$-partite systems
Mengying Hu, Ting Gao, Fengli Yan
- Year
- 2024
- Journal
- arXiv preprint
- DOI
- arXiv:2410.04331
- arXiv
- 2410.04331
A set of orthogonal states possesses the strongest quantum nonlocality if only a trivial orthogonality-preserving positive operator-valued measure (POVM) can be performed for each bipartition of the subsystems. This concept originated from the strong quantum nonlocality proposed by Halder $et~al.$ [Phy. Rev. Lett. $\textbf{122}$, 040403 (2019)], which is a stronger manifestation of nonlocality based on locally indistinguishability and finds more efficient applications in quantum information hiding. However, demonstrating the triviality of orthogonality-preserving local measurements (OPLMs) is not straightforward. In this paper, we present a sufficient and necessary condition for trivial OPLMs in $N$-partite systems under certain conditions. By using our proposed condition, we deduce the minimum size of set with the strongest nonlocality in system $(\mathbb{C}^{3})^{\otimes N}$, where the genuinely entangled sets constructed in Ref. [Phys. Rev. A $\textbf{109}$, 022220 (2024)] achieve this value. As it is known that studying construction involving fewer states with strongest nonlocality contribute to reducing resource consumption in applications. Furthermore, we construct strongest nonlocal genuinely entangled sets in system $(\mathbb{C}^{d})^{\otimes N}~(d\geq4)$, which have a smaller size than the existing strongest nonlocal genuinely entangled sets as $N$ increases. Consequently, our results contribute to a better understanding of strongest nonlocality.
Open paper