Corollary 4.3, p. 13 — for injective A, bᵢ ≠ 0 and |Ax| = b solvable, PhaseCutMod has a unique rank-one solution whenever PhaseLift does

Let $A\in\mathbb C^{n\times p}$ be a measurement matrix with rows $a_i^*$ and $b\in\mathbb R^n$ a vector of amplitudes. Assume that

1. $A$ is injective;
2. $b_i\neq 0$ for all $i=1,\dots,n$;
3. the phase recovery problem (1) admits an exact solution: there is $x\in\mathbb C^p$ with $|Ax|=b$.

Consider the two semidefinite relaxations

with $M=\operatorname{diag}(b)(\mathbf I-AA^\dagger)\operatorname{diag}(b)$ and $B=\operatorname{diag}(b)A^{\dagger *}A^\dagger\operatorname{diag}(b)$. If PhaseLift is tight, i.e. has a unique optimal solution and that solution has rank one, then PhaseCutMod is tight as well.

This is the paper's central noiseless result: the classical MaxCut-type relaxation, refined to PhaseCutMod, recovers the signal whenever PhaseLift does, so the exact-recovery guarantees known for PhaseLift transfer to it.

Formalization Note "Tight" is the corollary's own gloss, "has a unique rank one solution", applied to the optimal set of each program: the set of minimizers is a singleton $\{X\}$ with $\operatorname{Rank}(X)=1$. The gloss on p. 10 for PhaseLift adds "with leading eigenvector $x$"; that describes the known solution and is not part of the corollary's hypothesis, so it is not encoded. Objectives are real parts of traces; $\operatorname{Tr}(a_ia_i^*X)$ is $(AXA^*)_{ii}$; $A^\dagger$ is $(A^*A)^{-1}A^*$, the pseudoinverse under hypothesis 1.