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Global Completability with Applications to Self-Consistent Quantum Tomography

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Abstract

Let \({{p}_{1},\ldots, {p}_{N} \in \mathbb{R}^D}\) be unknown vectors and let \({\Omega \subseteq \{1,\ldots,N\}^{2}}\). 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 \({{p}_{1}, \ldots, {p}_{N}}\) uniquely? Here we derive a necessary and sufficient condition for the uniqueness of \({{p}_{1}, \ldots,{p}_{N}}\) (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 \({{p}_{1}, \ldots,{p}_{N} \in \mathbb{R}^{D}}\) the completability of \({{p}_{1}, \ldots,{p}_{N}}\) 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, matrix factorization techniques and self-consistent quantum tomography.

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  1. Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Cyril Jakob Stark

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Stark, C.J. Global Completability with Applications to Self-Consistent Quantum Tomography. Commun. Math. Phys. 348, 1–25 (2016). https://doi.org/10.1007/s00220-016-2760-2

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