Data processing for quantum mutual information ($I(A:B)\ge I(A:B\prime )$)

A local quantum channel on either subsystem cannot increase quantum mutual information. Indeed, apply the data-processing inequality for quantum relative entropy to $D({\rho }_{AB}\Vert {\rho }_A\otimes {\rho }_B)$; the channel maps the product of marginals to the product of the new marginals.

Let a uniformly random bit $A$ be copied into a three-state classical register $B$ with probability $\epsilon$, and otherwise replaced in $B$ by a flagged erasure independent of $A$. Then $I(A:B)=\epsilon$ 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.

For a channel on $B$, use a Stinespring dilation $B\to B^{\prime }E$. Isometry invariance gives $I(A:B)=I(A:B^{\prime }E)$, so the loss on discarding $E$ is $I(A:B)-I(A:B^{\prime })=I(A:E\mid B^{\prime })$. This quantum conditional mutual information is nonnegative by Strong subadditivity of Von Neumann entropy. The identity holds for arbitrary mixed inputs and identifies the correlations lost to the discarded environment.