# Frame Phase-retrievability and Exact phase-retrievable frames

## TL;DR

This paper studies the properties and existence of exact phase-retrievable frames in finite-dimensional Hilbert spaces, introducing concepts like PR-redundancy and analyzing maximal phase-retrievable subspaces.

## Contribution

It proves the existence of exact phase-retrievable frames for certain lengths and introduces the concept of PR-redundancy, extending phase-retrievability analysis to subspaces.

## Key findings

- Exact phase-retrievable frames exist for lengths between 2n-1 and n(n+1)/2 in real Hilbert spaces.
- Maximal PR-subspaces can have different dimensions, with characterizations based on support size.
- In basis cases, PR-subspace dimension relates to support size and maximality conditions.

## Abstract

An exact phase-retrievable frame $\{f_{i}\}_{i}^{N}$ for an $n$-dimensional Hilbert space is a phase-retrievable frame that fails to be phase-retrievable if any one element is removed from the frame. Such a frame could have different lengths. We shall prove that for the real Hilbert space case, exact phase-retrievable frame of length $N$ exists for every $2n-1\leq N\leq n(n+1)/2$. For arbitrary frames we introduce the concept of redundancy with respect to its phase-retrievability and the concept of frames with exact PR-redundancy. We investigate the phase-retrievability by studying its maximal phase-retrievable subspaces with respect to a given frame which is not necessarily phase-retrievable. These maximal PR-subspaces could have different dimensions. We are able to identify the one with the largest dimension, which can be considered as a generalization of the characterization for phase-retrievable frames. In the basis case, we prove that if $M$ is a $k$-dimensional PR-subspace, then $|supp(x)| \geq k$ for every nonzero vector $x\in M$. Moreover, if $1\leq k< [(n+1)/2]$, then a $k$-dimensional PR-subspace is maximal if and only if there exists a vector $x\in M$ such that $|supp(x) | = k$.