Phase retrieval at N = 4M − 5: refined injectivity probability

Determine for complex A ∈ C^{(4M−5)×M} drawn with random image uniformly from the Grassmannian whether the map x mod T → |Ax|^2 is injective with probability p_M strictly less than 1 for all M, and show that lim_{M→∞} p_M = 0.

Background

Originally, N ≥ 4M − 4 measurements were conjectured necessary and generically sufficient for complex phase retrieval; the necessity was refuted by an explicit injective example at N = 11, M = 4. A refined probabilistic conjecture posits that at N = 4M − 5, injectivity holds only with bounded-away-from-one probability, vanishing as M grows.

Resolving this would sharpen the boundary between necessary and sufficient measurement counts for generic complex phase retrieval.

References

Cynthia stated a refined version of the conjecture that still remains open (see~\url{https://dustingmixon.wordpress.com/2015/07/08/conjectures-from-sampta/}) Let N=4M-5 and draw A\inC{N\times M} at random (here the important thing is for $ \mathrm{im}(A) $ to be drawn uniformly from the Grassmannian of $ M $-dimensional subspaces of $ \mathbb{C}{4M-5}$, it can e.g. be with iid gaussian entries). Let p_M be the probability that the mapping $ x \bmod \mathbb{T} \mapsto |Ax|2 $ is injective. (a) $ p_M < 1 $ for all $ M $. (b) $ \displaystyle\lim_{M \to \infty} p_M = 0 $.

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture, Section “Injectivity and Stability of Phase Retrieval (ASB)” (Entry 10)