# Asymptotically Optimal Multi-Paving

## TL;DR

This paper extends techniques from the Kadison-Singer problem to demonstrate the existence of non-trivial pavings for collections of zero diagonal Hermitian matrices, providing asymptotic estimates and applications to commutator representations.

## Contribution

It develops a new technique based on Marcus, Spielman, and Srivastava's approach to show asymptotically optimal pavings for collections of matrices with zero diagonal, improving bounds and applications.

## Key findings

- Existence of pavings with r ≤ 18kε^{-2} for collections of matrices.
- Asymptotic estimates for paving general zero diagonal matrices.
- Simplified proof with better estimates for a theorem on commutator representations.

## Abstract

Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper, we develop a technique extending the arguments of Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem to show the existence of non-trivial pavings for collections of matrices. We show that given zero diagonal Hermitian contractions $A^{(1)}, \cdots, A^{(k)} \in M_n(\mathbb{C})$ and $\epsilon > 0$, one may find a paving $X_1 \amalg \cdots \amalg X_r = [n]$ where $r \leq 18k\epsilon^{-2}$ such that, \[\lambda_{max} (P_{X_i} A^{(j)} P_{X_i}) < \epsilon, \quad i \in [r], \, j \in [k].\] As a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices; zero diagonal contractions can be $(O(\epsilon^{-2}),\epsilon)$ paved. As an application, we give a simplified proof wth slightly better estimates of a theorem of Johnson, Ozawa and Schechtman concerning commutator representations of zero trace matrices.