{c}

If $U,V,X$ are independent random variables in $\mathbb F_2^n$, then
$$
d_R(U\mid U+V;X)
\leq\frac12\bigl(d_R(U;X)+d_R(V;X)+d_R(U;V)\bigr).
$$
Indeed, <conditioning reduces entropy> and independence give
$$
d_R(U\mid U+V;X)
\leq H(U+X)-\frac12H(U)-\frac12H(V)
+\frac12H(U+V)-\frac12H(X).
$$
In $\mathbb F_2^n$, $V=U+(U+V)$, so <conditional entropy under a deterministic change of variables> shows that the left-hand side is unchanged when $U$ and $V$ are exchanged. Averaging the displayed bound with its exchanged version gives the result.


Conditioned entropic Ruzsa distance of a summand

If $f(x,z)$ is a <bijection> of $x$ for every fixed $z$, then
$$
H(f(X,Z)\mid Z)=H(X\mid Z).
$$
In particular, translating a random variable in a finite group by a value determined by the conditioning variable does not change its conditional entropy.


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 Incoming links: Conditional entropy under a deterministic change of variables