Bargmann 不变量不足以完全区分局部酉轨道

中文摘要 AI 辅助

由二分密度算子及其两个提升边缘构成的 Bargmann 不变量是局部酉共轭的多项式不变量。我们确定了这些不变量所编码的精确信息。每当一个子系统是量子比特时，普通的边缘词族决定了部分转置的完整谱，从而决定该状态是否具有正部分转置性质。在 $2\otimes2$ 和 $2\otimes3$ 系统中，这给出了完全可分性判据。对于两量子比特系统，一个有限子族还能区分局部酉轨道，并且一个有限扩展生成多项式不变量环。这三个任务在量子比特-量子三态系统中已经出现分歧：我们展示了满秩、局部最大混合态，它们在所有普通边缘词不变量上一致，但具有不同的算子-Schmidt 秩，同时还有一个四次相关不变量将它们区分开。当两个局部维度都至少为 3 时，在局部最大混合扇区上的类似坍缩产生了等谱对，其中一个态是可分离态，另一个是具有负部分转置的纠缠态。因此，普通 Bargmann 代数与完整局部酉不变量环重合当且仅当两个子系统都是量子比特。缺失的数据是几何性的：它们编码了全局本征空间相对于张量积分解的位置。

英文摘要

Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In $2\otimes2$ and $2\otimes3$ systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition

发表机构

• School of Mathematical Sciences, Hangzhou Dianzi University（杭州电子科技大学数学科学学院）

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14 pages