Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank

We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d x d matrices A(1),..., A(n) each with parallel to A(i)parallel to(op) <= 1 and rank at most n/log(3) n, one can efficiently find +/- 1 signs x(1),..., x(n) such that their signed sum has spectral norm. parallel to Sigma(n)(i=1) xiAi parallel to(op) = O(root n). This result also implies a logn Omega(log logn) qubit lower bound for quantum random access codes encoding n classical bits with advantage >> 1/root n. Our proof uses the recent refinement of the non-commutative Khintchine inequality in [Bandeira, Boedihardjo, van Handel, 2021] for random matrices with correlated Gaussian entries.