Lemma: Conditional-entropy continuity

If $\dim {\mathcal{H}}_B=d$ and $d_{\mathrm{tr}}({\rho }_{AB},{\sigma }_{AB})\le \epsilon \le 1$, then

Proof. We give the finite-dimensional argument corresponding to Winter's continuity bound Winter. Let $x=d_{\mathrm{tr}}(\rho ,\sigma )$. If $x=0$, the conclusion is immediate. Otherwise the Jordan decomposition gives states ${\rho }_{+}$ and ${\rho }_{-}$ such that

Consequently,

For a binary mixture with weights $1-t,t$, conditional entropy satisfies

Indeed, attaching a classical mixture flag $Z$ identifies the difference with $I(B:Z|A)$. Its lower bound is strong subadditivity LiebRuskai. Its upper bound follows from $I(B:Z|A)=S(Z|A)-S(Z|AB)\le H(Z)$, since classical conditional entropy conditioned on a quantum system is nonnegative.

Apply this mixture inequality to the two presentations of $\omega$ with $t=x/(1+x)$. The difference between their mixture gaps has absolute value at most $h_2(t)$. Since $-{\log }_{2}d\le S(B|A)\le {\log }_{2}d$, we obtain

The right-hand side is increasing for $0\le x\le 1$, so replacing $x$ by $\epsilon$ proves the assertion.