Orthogonalization of Positive Operator Valued Measures

TL;DR

This paper proves that nearly orthogonal POVMs on a Hilbert space can be approximated by truly orthogonal POVMs within the same algebra, extending finite-dimensional results to infinite dimensions with optimal linear bounds.

Contribution

It generalizes previous finite-dimensional orthogonalization results for POVMs to infinite-dimensional settings with precise linear dependence bounds.

Findings

Nearly orthogonal POVMs are close to orthogonal POVMs in the same algebra.

Extension of finite-dimensional results to infinite-dimensional Hilbert spaces.

Achieved linear dependence bounds, proven to be optimal.

Abstract

We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe-Vidick and Ji-Natarajan-Vidick-Wright-Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal. We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra.