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Abstract：
For a quantum system with a d-dimensional Hilbert space, suppose a pure state |ψ⟩ is subjected to a complete orthogonal measurement. The measurement effectively maps |ψ⟩ to a point (p1,…,pd) in the appropriate probability simplex. It is a known fact-which depends crucially on the complex nature of the system\'s Hilbert space-that if |ψ⟩ is distributed uniformly over the unit sphere, then the resulting ordered set (p1,…,pd) is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional to dp1⋯dpd-1. In this paper we ask whether there is some foundational significance to this uniform measure. In particular, we ask whether it is the optimal measure for the transmission of information from a preparation to a measurement in some suitably defined scenario. We identify a scenario in which this is indeed the case, but our results suggest that an underlying real-Hilbert-space structure would be needed to realize the optimization in a natural way.
摘要：
对于具有d维希尔伯特空间的量子系统，假设纯态|进行完全正交测量。测量结果有效地将|Φ²映射到一个点(p1，...，pd)在适当的概率单纯形中。这是一个已知的事实-这在很大程度上取决于系统的希尔伯特空间的复杂性质-如果|Φ在单位球面上均匀分布，然后得到有序集(p1,...,pd)在概率单纯形上均匀分布；也就是说，单纯形上的所得度量与dp1_dpd-1成正比。在本文中，我们询问这种统一度量是否有一些基本意义。特别是,我们问，在一些适当定义的情况下，它是否是从准备到测量的信息传输的最佳措施。我们确定了一个确实如此的场景，但是我们的结果表明，需要一个潜在的真实希尔伯特空间结构来以自然的方式实现优化。
