发表机构
College of Science, Zhejiang University of Science and Technology; School of Mathematical Sciences, Hangzhou Dianzi University（浙江科技学院理学院; 杭州电子科技大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立量子多态在保对易操作下的精确资源转换理论，发现量子比特转换由三阶Bargmann不变量决定手性，并揭示检测阶数2与转换阶数3的分离，高维则结构刚性。
AI 中文摘要
量子多态的基础无关相干性由其组成态是否对易决定，对于量子比特，仅通过成对态重叠即可检测。决定其确定性变换的规律此前尚属开放问题。我们给出了在常见保对易操作（CPOs）下量子多态的精确资源转换理论。对于量子比特，成对Bargmann不变量决定唯一候选信道的奇异值，而单个三阶Bargmann相位决定其取向。完全正定性进而给出精确转换的充要条件。该条件识别出一个手性敏感壳层$\mathcal C_{\rm chir}$，在该壳层内，具有相同成对几何的目标态仅因三次不变量的符号不同而可能在可转换性上有所差异。这产生了阶数分离，$k_{\rm detect}=2<3=k_{\rm convert}$，适用于任意大小$n\geqslant3$的量子比特多态。手性敏感壳层恰好占据奇异值单纯形的三分之一，而镜像反射按体积阻碍了三分之二的可允许保取向变换。相比之下，对于$d\geqslant3$，结构是刚性的：每个有资源的转换要么通过各向同性退极化信道实现，要么通过转置退极化信道实现。可转换性随后由阶数至多$d^2$的中心Bargmann不变量决定。
英文摘要
Basis-independent coherence of a quantum multistate is fixed by whether its constituent states commute, and for qubits it is detected by pairwise state overlaps alone. What governs its deterministic transformation has been open. We give the exact resource-conversion theory of quantum multistates under common commutativity-preserving operations (CPOs). For qubits, pairwise Bargmann invariants determine the singular values of the unique candidate channel, while a single third-order Bargmann phase determines its orientation. Complete positivity then yields a necessary and sufficient criterion for exact conversion. The criterion identifies a chirality-sensitive shell $\mathcal C_{\rm chir}$, within which targets with identical pairwise geometry can differ in convertibility solely by the sign of their cubic invariant. This yields an order separation, $k_{\rm detect}=2<3=k_{\rm convert}$, for qubit multistates of every size $n\geqslant3$. The chirality-sensitive shell occupies exactly one third of the singular-value simplex, while mirror reflection obstructs two thirds of the admissible orientation-preserving transformations by volume. For $d\geqslant3$, by contrast, the structure is rigid: every resourceful conversion is realized by either an isotropic depolarizing or a transpose-depolarizing channel. Convertibility is then decided by centered Bargmann invariants of degree at most $d^2$.
Comments11 pages, 4 figures
