全局可完备性及其在自洽量子层析成像中的应用
摘要
设 p_1, ..., p_N \in R^D 为未知向量，且 Omega \subseteq {1,...,N}^{\times 2}。假设对于所有 (i,j) \in Omega，内积 p_i^T p_j 是固定的。这些内积约束（在所有向量同时旋转的意义下）是否能唯一确定 p_1, ..., p_N？在此，我们推导了 p_1, ..., p_N 唯一性（即全局可完备性）的充要条件，该条件适用于一大类具有实际意义的集合 Omega。此外，给定 Omega，我们证明了全局可完备性的条件是通用的，即对于几乎所有的向量 p_1, ..., p_N \in R^D，p_1, ..., p_N 的可完备性仅取决于 Omega，而不取决于 (i,j) \in Omega 对应的 p_i^T p_j 的具体数值。本工作的动机源于实际考量，即自洽量子层析成像。
引用
@article{arxiv.1209.6499,
title = {Global Completability with Applications to Self-Consistent Quantum Tomography},
author = {Cyril Stark},
journal= {arXiv preprint arXiv:1209.6499},
year = {2013}
}
备注
In the new version we focus entirely on the geometric problem described in the abstract. Proofs are more detailed and hopefully more transparent. To make contact with the literature on related questions, we talk about completability instead of rigidity. However, we had to remove the discussion of projective non-degenerate measurements. This investigation is better embedded in another manuscript