Four Deviations Suffice for Rank 1 Matrices

We prove a matrix discrepancy bound that strengthens the famous Kadison-Singer result of Marcus, Spielman, and Srivastava. Consider any independent scalar random variables $\xi_1, \ldots, \xi_n$ with finite support, e.g. $\{ \pm 1 \}$ or $\{ 0,1 \}$-valued random variables, or some combination thereof. Let $u_1, \dots, u_n \in \mathbb{C}^m$ and $$ \sigma^2 = \left\| \sum_{i=1}^n \text{Var}[ \xi_i ] (u_i u_i^{*})^2 \right\|. $$ Then there exists a choice of outcomes $\varepsilon_1,\ldots,\varepsilon_n$ in the support of $\xi_1, \ldots, \xi_n$ s.t. $$ \left \|\sum_{i=1}^n \mathbb{E} [ \xi_i] u_i u_i^* - \sum_{i=1}^n \varepsilon_i u_i u_i^* \right \| \leq 4 \sigma. $$ A simple consequence of our result is an improvement of a Lyapunov-type theorem of Akemann and Weaver.