# Mutual Information after Applying Random Unitary Matrix

Let $$\mathbf{U}$$ be a random unitary matrix and $$\mathbf{z}$$ be a random i.i.d complex Gaussian vector (unitary invariant). Assume that the following relation is satisfied: \begin{align} \mathbf{y}=\mathbf{U}\mathbf{s}+\mathbf{z}. \end{align} Are the following relations hold for the mutual information between $$\mathbf{s}$$ and $$\mathbf{y}$$? \begin{align} I(\mathbf{s};\mathbf{y})&=I(\mathbf{s};\mathbf{U}^{\mathrm{H}}\mathbf{y})\\ &=I(\mathbf{s};\mathbf{s}+\mathbf{U}^{\mathrm{H}}\mathbf{z})\\ &=I(\mathbf{s};\mathbf{s}+\mathbf{z})\\ &=h(\mathbf{s}+\mathbf{z})-h(\mathbf{z}), \end{align} where $$(.)^{\mathrm{H}}$$ is a hermitian opertor.

• This relationship is clearly true for deterministic $\mathbf{U}$. May 13, 2021 at 23:40

No, this is not correct. Consider as a counterexample the case that $$s$$ can take only two values, a unit vector $$e$$ or minus $$e$$. Since $$s$$ is rotated randomly to construct $$y=Us+z$$, knowledge of $$s$$ gives you no information on $$y$$, so the mutual information $$I(s,y)=0$$.
On the other hand, knowledge of $$s$$ does give you information on the sum $$s+z=\pm e+z$$, so $$I(s,s+z)\neq 0$$, contradicting the third equality in the OP.
The error appears already in the first equality: $$I(s,y)=I(s,U_0y)$$ for any given unitary $$U_0$$, but this is not the same unitary as in the construction $$y=Us+z$$, since that $$U$$ is unknown.