AI 中文总结
本文研究不可扩展稳定子基（USB），构造了四量子比特及奇素维度三 qudit 系统的 USB，证明其存在的最小规模，并对比了 USB 与不可扩展乘积基（UPB）的资源理论性质。
AI 中文摘要
我们研究一组不完备的正交稳定子态，这些态无法再添加一个与所有成员正交的稳定子态。这类集合是不可扩展乘积基（UPB）的稳定子类似物，因此我们称之为不可扩展稳定子基（USB）。利用素局域维度下n个 qudit 的 Pauli 群所基于的辛几何，我们明确构造了四量子比特系统和奇素局域维度下三个 qudit 系统的 USB。此外，我们证明这些分别是存在此类基的最小量子比特数和 qudit 数，且对于所有n≥4的量子比特系统以及所有n≥3的奇素维度 qudit 系统，均存在 USB。最后，我们比较了 USB 与 UPB 的资源理论方面，证明与 UPB 的情况类似，每个不可扩展稳定子集的正交补都是无稳定子子空间，其归一化投影算子必然是 magic 的；在奇素维度下它是 bound magic 的，但在量子比特中不一定是。我们还证明，仅 USB 的不可扩展性不会对稳定子操作的区分造成统一的定量阻碍。
英文摘要
We study incomplete sets of orthogonal stabiliser states that cannot be extended by a further stabiliser state orthogonal to all of its members. Such sets are the stabiliser analogue of unextendible product bases (UPBs), and we thus call them unextendible stabiliser bases (USBs). Leveraging the symplectic geometry underlying the $n$-qudit Pauli group in prime local dimension, we explicitly construct USBs for systems of four qubits and three qudits of odd prime local dimension. Moreover, we show that these are the respective minimal qubit, respectively qudit numbers for which such bases exist, and that USBs exist for all $n\geq4$ qubit and for all $n\geq3$ odd-prime-dimensional qudit systems. Finally, we compare the resource-theoretic aspects of USBs with those of UPBs. We establish that, analogous to the case of UPBs, the orthogonal complement of every unextendible stabiliser set is a stabiliser-free subspace, and its normalised projector is necessarily magic; moreover, it is bound magic in odd prime dimension, yet need not be for qubits. We also show that USB unextendibility alone imposes no uniform quantitative obstruction to discrimination by stabiliser operations.
Comments4+18 pages, 2 tables; comments welcome
