Theorem: Exact local entropy costs and cut additivity

Use natural logarithms and $S_2(\rho )=-\log \mathrm{Tr}({\rho }^2)$. The one-endpoint entropy of pair $e$ is exactly $w_e$. For every vertex bipartition $A\mid A^c$,

Proof. The reduced pair state has eigenvalues $(1-z_e){z_e}^r$, $r\ge 0$, so

Thus its Rényi-2 entropy is $w_e$.

A pair wholly on one side of the bipartition is a pure tensor factor on that side and contributes zero. A crossing pair contributes its one-endpoint reduced state. These factors are independent tensor factors. Their purities multiply, so their $S_2$ entropies add.