Consider the discrete random variables $X,Y,Z,W$, where $Y$ and $W$ are independent. Let $Y = f(X)$, for some function $f(\cdot)$, $Z=Y+W$, and $X \rightarrow Y \rightarrow Z$ be a Markov chain.

By definition, the mutual information satisfies $I(Y;Z) = H(Y) - H(Y|Z) = H(Z) - H(Z|Y)$.

I know that because $Z=Y+W$ and $Y$ and $W$ are independent, then $H(Z|Y) = H(Y+W|Y) = H(W|Y) = H(W)$.

However, I found someone claiming in a paper that $H(Y|Z) = H(Z-W|Z) = H(W|Z) = H(W)$. I cannot understand this because $W$ and $Z$ are correlated. So why is it true that $H(W|Z) = H(W)$? Is it because of the Markov chain assumption?

Any insight would be greatly appreciated! Thanks for reading me.