Proposition: Dimension-priced entropy comparison

For normalized pure vectors $\psi ,\varphi$ with $\Vert \psi -\varphi \Vert \le \epsilon \le 1$,

Proof. The trace distance of the pure states is at most their vector distance. Partial trace contracts trace distance. Apply the sharp finite-dimensional entropy continuity inequality to the smaller side of the pure bipartition Audenaert. The capped modulus is its maximum over distances at most $\epsilon$.