= Solution
Combining parts (iii) and (iv), then multiplying by $1+\varepsilon$, gives
$$
H(A|B)_\sigma-H(A|B)_\rho
\leq\varepsilon\bigl(H(A|B)_{\Delta'}-H(A|B)_\Delta\bigr)
+(1+\varepsilon)H\!\left(\frac{\varepsilon}{1+\varepsilon}\right).
$$
The <dimension bound for quantum conditional entropy> is
$$
-\log d_A\leq H(A|B)_\tau\leq\log d_A.
$$
Indeed, <Subadditivity of Von Neumann entropy> gives $H(A|B)_\tau\leq S(\tau_A)\leq\log d_A$, while the <Araki–Lieb inequality> gives $H(A|B)_\tau\geq-S(\tau_A)\geq-\log d_A$. Hence
$$
H(A|B)_\sigma-H(A|B)_\rho
\leq2\varepsilon\log d_A
+(1+\varepsilon)H\!\left(\frac{\varepsilon}{1+\varepsilon}\right).
$$
Interchanging $\rho$ and $\sigma$ proves the <continuity bound for quantum conditional entropy>:
$$
\boxed{|H(A|B)_\rho-H(A|B)_\sigma|
\leq2\varepsilon\log d_A
+(1+\varepsilon)H\!\left(\frac{\varepsilon}{1+\varepsilon}\right)}.
$$
Solved by gpt-5.6-sol high.
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