Proposition: Elementary conditional consequences

If a pure state satisfies $S(A)\le C|{\partial }_{\Lambda }A|$, then:

1. for every $\alpha >1$, including the limiting min-entropy case,

   $$S_{\alpha }(A)\le S(A)\le C|{\partial }_{\Lambda }A|;$$
2. $$I(A:A^c)=2S(A)\le 2C|{\partial }_{\Lambda }A|;$$

3. for disjoint $X,Y$,

   $$I(X:Y)\le 2\min \{S(X),S(Y)\}.$$

   These do not give a distance-decay theorem, an $\alpha <1$ Rényi bound, or a topological correction.

Proof. For eigenvalues $p_i$ and $\alpha >1$, Jensen's inequality gives

Taking logarithms and dividing by $1-\alpha <0$ proves the Rényi assertion. The limiting case follows by continuity. Purity proves the complementary mutual-information identity. The last assertion follows from the Araki–Lieb inequality.