and MMSE for SMV complex CS with complex measurement matrices are the same as MMV-1 with $J=2$. Note that in the free energy expression  of MMV-1, the MSE, $D=E$ , is the average MSE of the $J$ entries of $\s_l$. Therefore, in this complex CS setting, $D$ is the average MSE of the real and imaginary parts of the signal entries.

Proof of Lemma \[lemma:covIsSame\] {#sec:proof}
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In this section, we show that the quantity $\mathbb{X_{\mu}}$ is the same for MMV-1 and MMV-2. Moreover, we show that complex SMV with a complex measurement matrix also yields the same $\mathbb{X_{\mu}}$ with $J=2$.

First, we re-write  in the vector form $$\label{eq:XMuVector}
\mathbb{X}_{\mu}\!=\!\mathbb{E}_{\v_{\mu}}\!\left[\operatorname{e}^{-\frac{1}{2\Delta}{\sum_{j=1}^{J}}{\sum_{a=1}^{n}}(v_{\mu j}^a)^2}\right]\!
=\!\mathbb{E}_{\v_{\mu}}\!\left[\operatorname{e}^{-\frac{1}{2\Delta}\v_{\mu}^T\v_{\mu}}\right],$$ where $\v_{\mu}=[v_{\mu 1}^1,...,v_{\mu 1}^a,...,v_{\mu J}^1$, $...,v_{\mu J}^n]^T$ and $v_{\mu j}^a$ is given in . In order to calculate the expectation w.r.t. $\v_{\mu}$ in , we calculate the distribution of $\v_{\mu}$, which is approximated by a Gaussian distribution, due to the central limit theorem. The mean is $\mathbb{E}_{\F,\z}[v_{\mu j}^a]=0$.

We now calculate the covariance matrix, $\G_{\mu}=\mathbb{E}[\v_{\mu}\v_{\mu}^T]$. The matrix is separated into $J\times J$ blocks of size $n\times n$, as shown in Fig. \[fig.cov\]. The main diagonal of $\G_{\mu}$ consists of entries $w_1=\mathbb{E}_{\F,\z}[(v_{\mu j}^a)^2]$. The entries in the blocks along 