be written as real-valued MMV, $$\label{eq:complexRealMat}
    \y^{\mathcal{R}}=\F^{\mathcal{R}}\s^{\mathcal{R}}+\z^{\mathcal{R}}\ \text{and}\     \y^{\mathcal{I}}=\F^{\mathcal{R}}\s^{\mathcal{I}}+\z^{\mathcal{I}},$$ where $\s^R$ and $\s^I$ are jointly sparse and follow . This formulation  fits into MMV-2 for $J=2$. Hence, we can obtain the MMSE according to .[^9]

[**Complex measurement matrix:**]{} Consider a complex $\F^{\mathcal{C}}=\F^{\mathcal{R}}+i\F^{\mathcal{I}}\in\mathbb{C}^{M\times N}$ with entries $F_{\mu l}^{\mathcal{R}}, F_{\mu l}^{\mathcal{I}}\sim \mathcal{N}(0,\frac{1}{2N})$. Expanding out the complex channel, $\y^{\mathcal{C}}=\F^{\mathcal{C}}\s^{\mathcal{C}}+\z^{\mathcal{C}}$, we obtain the equivalent real-valued SMV channel, $$\label{eq.realComplexChannel}
  \begin{bmatrix}
    \y^{\mathcal{R}} \\
    \y^{\mathcal{I}}
  \end{bmatrix}
=
  \begin{bmatrix}
    \F^{\mathcal{R}} & -\F^{\mathcal{I}}  \\
    \F^{\mathcal{I}}  & \F^{\mathcal{R}}
  \end{bmatrix}
    \begin{bmatrix}
    \s^{\mathcal{R}} \\
    \s^{\mathcal{I}}
  \end{bmatrix}
  +
    \begin{bmatrix}
    \z^{\mathcal{R}} \\
    \z^{\mathcal{I}}
  \end{bmatrix}.$$

We re-arrange  as follows, $$\label{eq:complexMatRearrange}
\underbrace{\begin{bmatrix}
    \y^{\mathcal{R}} \\
    \y^{\mathcal{I}}
  \end{bmatrix}}_{\overline{\y}}
\!=\!
\underbrace{\begin{bmatrix}
    \F_{:,1}^{\mathcal{R}},-\F_{:,1}^{\mathcal{I}},...,\F_{:,N}^{\mathcal{R}}, -\F_{:,N}^{\mathcal{I}} \\
    \F_{:,1}^{\mathcal{I}},\ \ \F_{:,1}^{\mathcal{R}},...,\F_{:,N}^{\mathcal{I}},\ \  \F_{:,N}^{\mathcal{R}}
  \end{bmatrix}}_{\overline{\F}}
    \underbrace{\begin{bmatrix}
    s_1^{\mathcal{R}}\\
    s_1^{\mathcal{I}}\\
    \vdots\\
    s_N^{\mathcal{R}}\\
    s_N^{\mathcal{I}}
  \end{bmatrix}}_{\overline{\s}}
  \!+\!
    \underbrace{\begin{bmatrix}
    \z^{\mathcal{R}} \\
    \z^{\mathcal{I}}
  \end{bmatrix}}_{\overline{\z}},$$ where $\{:\}$ refers to all the rows. In the re-arranged channel , the measurement matrix $\overline{\F}$ consists of super symbols, $$\label{eq:SMV_F}
    \overline{\F}_{\mu l}=\left\{
                \begin{array}{ll}
                 &[F_{\mu l}^{\mathcal{R}},-F_{\mu l}^{\mathcal{I}}],\ \mu\in\{1,...,M\}\\
                 &[F_{\mu l}^{\mathcal{I}}, F_{\mu l}^{\mathcal{R}}],\ \mu\in\{M+1,...,2M\}
                \end{array}
            \right.,\\$$ and the signal $\overline{\s}$ consists of $\overline{\s}_{l}=\begin{bmatrix}
    s_l^{\mathcal{R}} \\
    s_l^{\mathcal{I}}
  \end{bmatrix},\ l\in\{1,...,N\}$. The measurements and noise are $\overline{\y}=\begin{bmatrix}
    \y^{\mathcal{R}} \\
    \y^{\mathcal{I}}
  \end{bmatrix}$ and $\overline{\z}=\begin{bmatrix}
    \z^{\mathcal{R}} \\
    \z^{\mathcal{I}}
  \end{bmatrix}$, respectively. Hence, $\overline{y}_{\mu}=\sum_{l=1}^N \overline{\F}_{\mu l}\overline{\s}_l+\overline{z}_{\mu},\ \mu\in\{1,...,2M\}$.

Section \[sec:proof\] shows that the free energy 