In quantum mechanics, states are described by density matrices. Though their probabilistic interpretation is rooted in ensemble theory, density matrices embody a known shortcoming. They do not completely express the physical realization of an ensemble. Conveniently, the outcome statistics of projective and positive operator-valued measurements do not depend on the ensemble realization, only on the density matrix. Here, we show how the geometric approach to quantum mechanics tracks ensemble realizations. We do so in two concrete cases of a finite-dimensional quantum system interacting with another one with (i) finite-dimensional Hilbert space, relevant for quantum thermodynamics, and (ii) infinite-dimensional Hilbert space, relevant for state-manipulation protocols.

中文翻译：

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超越密度矩阵：几何量子态

在量子力学中，状态由密度矩阵描述。尽管他们的概率解释植根于集成理论，但密度矩阵体现了一个已知的缺点。它们并不能完全表达合奏的物理实现。方便的是，投影和正算子值测量的结果统计不依赖于集成实现，只依赖于密度矩阵。在这里，我们展示了量子力学的几何方法如何跟踪集成实现。我们在有限维量子系统与另一个具有（i）有限维希尔伯特空间（与量子热力学相关）和（ii）无限维希尔伯特空间（与状态操纵协议）相互作用的两个具体案例中这样做。