A lower bound for the Balan--Jiang matrix problem

We prove the existence of a positive semidefinite matrix A ∈ Rn × n such that any decomposition into rank-1 matrices has to have factors with a large ¹-norm, more precisely ∑ₖ xₖ xₖ^*=A ∑ₖ \|xₖ\|²₁ ≥ c √n \|A\|₁, where c is independent of n. This provides a lower bound for the Balan--Jiang matrix problem. The construction is probabilistic.