Zero mutual information I have two random variables X and Y=X+N. What should be the condition such that I(X; Y)=0? Can anyone direct me towards the relevant references?

Just use the definition
$$
\operatorname{I}(X;Y) = 
  \int_{\mathcal Y} \int_{\mathcal X}
      {p_{(X,Y)}(x,y) \log{ \left(\frac{p_{(X,Y)}(x,y)}{p_X(x)\,p_Y(y)} \right) }
  } \; dx \,dy
$$
For it to be zero, you either need $p_{(X,Y)}(x, y)$ to be always zero, but then it wouldn't be a proper distribution, or the log part would need to be equal to zero. Recall that $\log(1) = 0$, so you need $p_{(X,Y)}(x, y) = p_X(x) \, p_Y(y)$, i.e. $X$ and $Y$ would need to be independent.
