从完整单周期端点数据中无法全局识别富比尼-施图迪几何
Global Non-Identifiability of Fubini-Study Geometry from Complete One-Period Endpoint Data
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中文总结 AI 辅助
该研究证明周期驱动有限维量子系统中，完整单周期端点数据无法全局确定周期平均富比尼-施图迪几何，明确了数据与动力学间的信息缺口。
中文摘要 AI 辅助
我们针对周期驱动的有限维量子系统，建立了一个全局最坏情况下的不可识别性定理。在无约束的光滑周期哈密顿量类上，我们证明，对于固定初态的周期平均富比尼-施图迪度量分量，一般无法从由所有起始时间和外部参数索引的精确单周期传播子中重构。因此，即使是完整的、按起始时间解析的端点数据，也不足以确定这一周期内的几何量。该阻碍被精确刻画：按起始时间索引的端点数据决定了单值性的共轭路径，但无法确定其特定的酉提升。在每个固定单值性的切片上，观测纤维恰好是由光滑参数相关的基于环生成的右轨道，这些环取值于单值性的逐点中心化子。富比尼-施图迪泛函在该纤维作用下不具有不变性，因此无法通过端点观测映射分解。一个显式的实解析两能级实例在交换的单生成元哈密顿量族中证明了该阻碍，因此既不需要非阿贝尔时间序，也不需要弗洛凯对数歧义。一个连续的哈密顿量族产生相同的按起始时间索引的单周期端点数据，同时产生不同的、且在无约束类中可任意分离的周期平均富比尼-施图迪几何。不可识别性在哈密顿量参数导数的任意规定均匀界下仍然存在。该结果是确定性的全局陈述，而非一般不可识别性或实验不可能的主张，它明确了完整单周期端点数据与完整周期内动力学之间的精确信息缺口：缺失的信息是观测到的单值性路径的酉提升，而非普通标量相位自由度。
英文摘要
We establish a global, worst-case non-identifiability theorem for periodically driven finite-dimensional quantum systems. On the unrestricted smooth periodic Hamiltonian class, we show that the period-averaged Fubini-Study metric component of a fixed initial state cannot, in general, be reconstructed from exact one-period propagators indexed by every starting time and external parameter. Thus, even complete starting-time-resolved endpoint data are insufficient to determine this intra-period geometric quantity. The obstruction is characterized exactly. The starting-time-indexed endpoint data determine the conjugation path of the monodromy, but not its particular unitary lift. On each fixed-monodromy slice, the observational fibres are precisely the right orbits generated by smooth parameter-dependent based loops taking values in the pointwise centralizer of the monodromy. The Fubini-Study functional is not invariant under this fibre action and therefore does not factor through the endpoint observation map. An explicit real-analytic two-level witness demonstrates the obstruction within a commuting, one-generator Hamiltonian family, so neither non-Abelian time ordering nor Floquet-logarithm ambiguity is required. A continuous family of Hamiltonians produces identical starting-time-indexed one-period endpoint data while yielding different, and on the unrestricted class arbitrarily separated, period-averaged Fubini-Study geometry. Non-identifiability persists under any prescribed uniform bound on the parameter derivative of the Hamiltonian. The result is a deterministic global statement, not a claim of generic non-identifiability or experimental impossibility. It identifies a precise information gap between complete one-period endpoint data and full intra-period dynamics: the missing information is the unitary lift of the observed monodromy path rather than ordinary scalar phase freedom.
