Global Completability with Applications to Self-Consistent Quantum Tomography

Abstract

Let p1,...,pN ∈ R[superscript D] be unknown vectors and let Ω ⊆ {1,...,N}[superscript 2]. Assume that the inner products p[superscript T][subscript i]p[subscript j] are fixed for all (i,j) ∈ Ω. Do these inner product constraints (up to simultaneous rotation of all vectors) determine p1,...,pN uniquely? Here we derive a necessary and sufficient condition for the uniqueness of p1,..., pN (i.e., global completability) which is applicable to a large class of practically relevant sets Ω. Moreover, given Ω, we show that the condition for global completability is universal in the sense that for almost all vectors p1,...,pN ∈ R[superscript D] the completability of p1,...,pN only depends on Ω and not on the specific values of p[superscript T][subscript i]p[subscript j] for (i,j) ∈ Ω. This work was motivated by practical considerations, namely, matrix factorization techniques and self-consistent quantum tomography.