标题： 量子计量学的几何不变量 显示英文标题

标题： Geometric Invariants of Quantum Metrology

摘要： 我们建立了一个之前未被探索的量子费舍尔信息矩阵(QFIM)的守恒定律，其表达如下；当QFIM由一个在对易下封闭的可观测量集合构造时，即一个李代数，QFIM的谱在由这些相同算子生成的幺正动力学下保持不变。 因此，每个李代数都赋予任何量子态一个固定的“计量灵敏度”预算——一种我们证明的内在资源，类似于干涉仪中的光学压缩，不能通过保持对称性的操作来放大。 乌尔曼曲率张量(UCT)自然继承相同的对称群，因此量子不相容性同样固定。 结果出现了一个计量学上的类似刘维尔定理的结论；统计距离、体积和曲率在由李代数生成的演化下保持不变。 我们讨论了这与经典最优标准的量子对应的关系。 这使得能够通过几何不变量在李代数层面高效地分类有用的量子态类别。 显示英文摘要

摘要： We establish a previously unexplored conservation law for the Quantum Fisher Information Matrix (QFIM) expressed as follows; when the QFIM is constructed from a set of observables closed under commutation, i.e., a Lie algebra, the spectrum of the QFIM is invariant under unitary dynamics generated by these same operators. Each Lie algebra therefore endows any quantum state with a fixed "budget" of metrological sensitivity -- an intrinsic resource that we show, like optical squeezing in interferometry, cannot be amplified by symmetry-preserving operations. The Uhlmann curvature tensor (UCT) naturally inherits the same symmetry group, and so quantum incompatibility is similarly fixed. As a result, a metrological analog to Liouville's theorem appears; statistical distances, volumes, and curvatures are invariant under the evolution generated by the Lie algebra. We discuss this as it relates to the quantum analogs of classical optimality criteria. This enables one to efficiently classify useful classes of quantum states at the level of Lie algebras through geometric invariants.