Worked solution: Marcus–Spielman–Srivastava proof via interlacing families (2013)

Kadison and Singer (1959) asked whether every pure state on the diagonal masa (maximal abelian subalgebra) of $B(\ell^2)$ extends uniquely to a pure state on all of $B(\ell^2)$. This is a question in operator algebra theory about how much information a "restricted" measurement determines about a full quantum-mechanical observable, and it resisted proof or disproof for decades.

Nik Weaver (2004) proved that the Kadison–Singer problem is logically equivalent to a family of finite-dimensional statements, the strongest of which is called $KS_2$: given any collection of vectors $v_1,\ldots,v_m$ in a finite-dimensional space $\mathbb{C}^d$ summing to the identity, $\sum_i v_iv_i^*=I_d$, with each vector short, $\|v_i\|^2\le\delta$, one can always split the index set into two parts $S_1,S_2$ so that the partial sums $\sum_{i\in S_j}v_iv_i^*$ are bounded away from the full identity. Marcus, Spielman and Srivastava (2013, published in the Annals of Mathematics in 2015) set out to prove exactly this statement $KS_2$.

This reduction is what makes the problem tractable: instead of reasoning about infinite-dimensional operator algebras and non-constructive objects like ultrafilters, the whole question becomes a concrete, checkable claim about splitting finitely many matrices — the object of study for the rest of this proof.