Asymptotically Optimal Multi-Paving

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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⁾,, A⁽ᵏ⁾ ∈ Mₙ(C) and ε> 0, one may find a paving X₁ Xᵣ = [n] where r ≤ 18kε⁻² such that, λmax (Pₓᵢ A⁽ʲ⁾ Pₓᵢ) < ε, i ∈ [r], j ∈ [k]. As a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices; zero diagonal contractions can be (O(ε⁻²),ε) 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.