Dgdrxrt

Suppose we have a function $f$,

$$
C = f(A,B),
$$

where $A$, $B$ and $C$ are random variables.

I notice that when the random variables are binary ($0, 1$) and $f$ is the XOR operation, we have the following identity:

$$
H(C|A) = H(B),
$$

where $H(B)$ is the entropy of $B$ and $H(C|A)$ is the conditional entropy of $C$ given $A$.

Obviously this is not true for a general $f$. What I am interested to know is, is there a set of conditions on $f$ and $A,B,C$, under which the identity above holds.

The following answer assumes that $A,B$ are independent, and that $A,B$ have full support on their respective domains (the latter is without loss of generality). For the general case, see the other answer.

Let's write your equation in a slightly different way:
$$
H(B) = H(f(A,B)|A) = operatorname*mathbbE_a sim A H(f(a,B)).
$$
Clearly $H(f(a,B)) leq H(B)$, with equality if and only if $f(a,b_1) neq f(a,b_2)$ whenever $b_1 neq b_2$. We deduce that $H(B) = H(f(A,B)|A)$ if and only if $f$ is injective in its second argument, i.e., for all $a$ and $b_1 neq b_2$, we have $f(a,b_1) neq f(a,b_2)$.

Note

beginalign
0&=H(C|A,B)\
&=H(A,B,C)-H(A,B)\
&=H(B|A,C)+H(C|A)+H(A)-H(A,B)quadtext(chain rule)\
&=H(B|A,C)+H(C|A)-H(B|A),
endalign

so $H(C|A)=H(B)$ is equivalently $H(B|A,C)+H(B)-H(B|A)=0$. Also note $H(B|A,C)ge 0$ and $H(B)ge H(B|A)$, your condition is equivalently $H(B|A,C)=0wedge H(B)=H(B|A)$.

For a human-readable explanation, $H(B|A,C)=0$ means $B$ is determined by $A$ and $C$, that is, for any fixed $a$ in the support of $A$, $f(a,b)$ as a function of $b$ with domain $bmid mathrmPrA=a, B=b>0$ is an injection. $H(B)=H(B|A)$ means $A$ and $B$ are independent of each other.