Optimality of the entrywise-bb1 sufficient condition for k-learnability

Determine whether the entrywise coherence bound in Theorem 1, namely the condition that |⟨ψ_i|ψ_j⟩| ≤ √((k−1)/((n−1)(n−k+1))) for all i ≠ j, is optimal for guaranteeing k-learnability of a multiset of n pure quantum states.

Background

The paper establishes two complementary universal criteria for k-learnability. Theorem 1 gives a sufficient condition based on the squared Frobenius norm of the Gram matrix and, as a consequence, an entrywise bound on the pairwise overlaps of the states. The authors prove that the Frobenius-norm condition is tight by constructing Gram matrices that cease to be k-learnable when the norm is increased slightly beyond the stated threshold.

The corresponding entrywise condition is obtained by bounding the Frobenius norm using the maximum pairwise overlap. The paper does not determine whether this derived entrywise threshold can be improved while still guaranteeing k-learnability, and explicitly identifies its optimality as unresolved.