= Postselection can increase conditional quantum mutual information
Let a uniformly random bit $A$ be copied into a three-state classical register $B$ with probability $\varepsilon$, and otherwise replaced in $B$ by a flagged erasure independent of $A$. Then $I(A:B)=\varepsilon$ bits. Projecting $B$ onto its two nonerased states and conditioning on success gives a perfectly correlated bit pair with mutual information one. The <postselection> branch is completely positive and trace decreasing, but renormalization after conditioning is not a deterministic <quantum channel>. This does not contradict <data processing for quantum mutual information>, which applies to trace-preserving operations or an unconditioned instrument including its outcome flag.
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