arXiv 1209.64992012-09quant-phcond-mat.mes-hall

全局可补性及其在自洽量子层析中的应用

Global Completability with Applications to Self-Consistent Quantum Tomography

Cyril Stark

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

摘要：

设p_1，.，p_N \in R_D是未知向量，并且令Ω \subseteq {1，.，N}^{ \times 2}。假设对于所有（i，j）\in Ω，内积p_i^T p_j是固定的。这些内积约束（直到所有向量的同时旋转）确定p_1，.，p_N唯一？这里我们得到了p_1，…，p_N（即，全局完备性），这适用于一个大类的实际相关的集合Ω。此外，给定Omega，我们证明了全局完备性的条件是普适的，即对于几乎所有向量p_1，...，p_N \in R^D p_1，.的完备性，p_N仅取决于Ω，而不取决于（i，j）\in Ω的p_i^T p_j$的特定值。这项工作的动机是实际的考虑，即自洽的量子层析成像。

英文摘要：

Let p_1, ..., p_N \in R^D be unknown vectors and let Omega \subseteq {1,...,N}^{\times 2}. Assume that the inner products p_i^T p_j are fixed for all (i,j) \in Omega. Do these inner product constraints (up to simultaneous rotation of all vectors) determine p_1, ..., p_N uniquely? Here we derive a necessary and sufficient condition for the uniqueness of p_1, ...,p_N (i.e., global completability) which is applicable to a large class of practically relevant sets Omega. Moreover, given Omega, we show that the condition for global completability is universal in the sense that for almost all vectors p_1, ...,p_N \in R^D the completability of p_1, ...,p_N only depends on Omega and not on the specific values of p_i^T p_j$ for (i,j) \in Omega. This work was motivated by practical considerations, namely, self-consistent quantum tomography.

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摘要：

英文摘要：

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