For classical-quantum states, the conditional entropy can change by at most epsilon log2(d_B-1) + h2(epsilon) under a trace-distance perturbation epsilon, and this bound cannot be improved.
A Tight Uniform Continuity Bound for Equivocation
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We prove a tight uniform continuity bound for the conditional Shannon entropy of discrete finitely supported random variables in terms of total variation distance.
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Optimal uniform continuity bound for conditional entropy of classical--quantum states
For classical-quantum states, the conditional entropy can change by at most epsilon log2(d_B-1) + h2(epsilon) under a trace-distance perturbation epsilon, and this bound cannot be improved.