Phase retrieval and norm retrieval

TL;DR

This paper explores the conditions under which phase retrieval properties transfer to complements in frames and introduces norm retrieval as a key concept linking phase retrieval to its orthogonal complements.

Contribution

It introduces norm retrieval, establishes its necessity for phase retrieval to pass to complements, and provides a comprehensive study of its relationship with phase retrieval.

Findings

Phase retrieval passes to Naimark complements under certain conditions.

Norm retrieval is necessary for phase retrieval to pass to orthogonal complements.

A frame yields phase retrieval iff its transformed frames yield norm retrieval for all invertible operators.

Abstract

Phase retrieval has become a very active area of research. We will classify when phase retrieval by Parseval frames passes to the Naimark complement and when phase retrieval by projections passes to the orthogonal complements. We introduce a new concept we call norm retrieval and show that this is what is necessary for passing phase retrieval to complements. This leads to a detailed study of norm retrieval and its relationship to phase retrieval. One fundamental result: a frame $\{\varphi_i\}_{i=1}^M$ yields phase retrieval if and only if $\{T\varphi_i\}_{i=1}^M$ yields norm retrieval for every invertible operator $T$.