large system limit, the quantity $\mathbb{X}_{\mu}$  is the same for both MMV-1 and MMV-2.

Lemma \[lemma:covIsSame\] is proved in Section \[sec:proof\]. Because of Lemma \[lemma:covIsSame\], the free energy expressions for MMV-1 and MMV-2 should be identical in the large system limit. We state the result as a theorem and the detailed derivations appear in the Appendix.

\[th:free\_energy\] For settings MMV-1 and MMV-2, the free energy expressions as functions of $E$ are identical in the large system limit and are given in .[^6]

$$\begin{aligned}
  \mathcal{F}(E)&=&-\frac{J}{2}R\l\{\log[2\pi(\Delta+E)]+\frac{\rho+\Delta}{E+\Delta}\r\}\!+\!\int\!P(\s_1)\! \int\! \log\! \l[ \int P(\x_1)\operatorname{e}^{-\frac{\widehat{Q}+\widehat{q}}{2}\x_1^T\x_1+\widehat{m}\x_1^T\s_1+
  \sqrt{\widehat{q}}\h^T\x_1}\!d\x_1\!\r]\!\mathcal{D}\h \ d\s_1\label{eq:free_energy3}\\
  &=&-\frac{J}{2}R\l\{\log[2\pi(\Delta+E)]+\frac{\Delta}{E+\Delta}\r\}+\frac{JR(1-\rho)}{2(R+E+\Delta)}+\rho\int \log \Bigg[ \rho \l(\frac{E+\Delta}{R+E+\Delta}\r)^{J/2}+\nonumber\\
  &\ &(1-\rho)\operatorname{e}^{-\frac{R}{2(E+\Delta)}\g^T\g}\Bigg]\mathcal{D}\g+(1-\rho)\int \log \l[ \rho \l(\frac{E+\Delta}{R+E+\Delta}\r)^{J/2}+(1-\rho)\operatorname{e}^{-\frac{R}{2(R+E+\Delta)}\h^T\h}\r]\mathcal{D}\h.\label{eq:free_energy4}\end{aligned}$$

**MMSE**: The $E$ that maximizes the free energy  the MMSE [@Krzakala2012probabilistic; @krzakala2012statistical; @Barbier2015]. After finding the $E_0$ that maximizes the free energy , we obtain the MMSE, $D_0=E_0$, in the large system limit.

The MMSE for MMV-1 and MMV-2 is the same for the same measurement rate $R$, noise variance $\Delta$, and number of signal vectors $J$.

[**Remark 1:**]{} As the reader can see from the proof of Lemma \[th:free\_energy\] in Section \[sec:proof\], the key reason that both MMV-1 and MMV-2 have an identical MMSE is that the entries in the super symbols $\s_l$ and $\x_l^{\cdot}$ are 