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 $d \times d$ matrices $A1,\ldots,An$ each with $|Ai|{\mathsf{op}} \leq 1$ and rank at most $n/\log³ n$, one can efficiently find $\pm 1$ signs $x1,\ldots,xn$ such that their signed sum has spectral norm $|\sum{i=1}ⁿ xi Ai|{\mathsf{op}} = O(\sqrt{n})$. This result also implies a $\log n - \Omega( \log \log n)$ qubit lower bound for quantum random access codes encoding $n$ classical bits with advantage $\gg 1/\sqrt{n}$. Our proof uses the recent refinement of the non-commutative Khintchine inequality in [Bandeira, Boedihardjo, van Handel, 2022] for random matrices with correlated Gaussian entries.