Proposition: Affine rigidity

At a fixed finite size, an affine, positive, normalized assignment on density operators that agrees with the diagonal on diagonal states is the Born diagonal assignment.

Proof. Every affine coordinate on the trace-one Hermitian hyperplane has the form $\mathrm{tr}(E_x\rho )$ for a Hermitian $E_x$; its affine constant can be absorbed into a multiple of the identity. Positivity on all states gives $E_x\ge 0$. Normalization gives ${\sum }_x{E}_x=I$.

Anchoring on basis states gives

For a positive matrix,

All off-diagonal entries vanish, and the only nonzero diagonal entry is the $x$-entry, equal to one. Thus $E_x=|x⟩⟨x|$.