Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank

TL;DR

This paper proves the matrix Spencer conjecture for matrices with poly-logarithmic rank, providing an efficient method to find signs that bound the spectral norm of their signed sum, with implications for quantum information theory.

Contribution

The paper offers a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank using recent non-commutative Khintchine inequalities, advancing understanding of matrix signings.

Findings

Efficiently finds signs with spectral norm O(√n) for matrices with poly-logarithmic rank.

Proves the matrix Spencer conjecture up to poly-logarithmic rank.

Derives a quantum lower bound for random access codes with classical bits.

Abstract

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, 2022] for random matrices with correlated Gaussian entries.