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 $A_1,\ldots,A_n$ each with $\|A_i\|_{\mathsf{op}} \leq 1$ and rank at most $n/\log^3 n$, one can efficiently find $\pm 1$ signs $x_1,\ldots,x_n$ such that their signed sum has spectral norm $\|\sum_{i=1}^n x_i A_i\|_{\mathsf{op}} = O(\sqrt{n})$. 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, 2021] for random matrices with correlated Gaussian entries.