Global Completability with Applications to Self-Consistent Quantum Tomography
Abstract
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.
Cite
@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}
}
Comments
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