physics method that can be used to analyze the MMSE and phase transition for inverse problems [@Tanaka2002; @GuoVerdu2005; @Montanari2006; @Krzakala2012probabilistic; @krzakala2012statistical; @MezardMontanariBook; @Barbier2015; @Lesieur2015]. Barbier and Krzakala [@Barbier2015] studied the MMSE for estimating superposition codes using replica analysis. In this paper, we extend the derivation in Barbier and Krzakala [@Barbier2015] to two related yet different MMV settings: ([*i*]{}) $J$ jointly sparse signals are measured by $J$ different dense matrices that are independent and identically distributed (i.i.d.), and ([*ii*]{}) $J$ jointly sparse signals are measured by $J$ identical i.i.d. matrices. We only consider dense i.i.d. Gaussian matrices in this work, while our analysis can be extended to other i.i.d. matrices easily.

We make several contributions in this paper. First, we obtain the information theoretic MMSE for the two MMV settings above under the Bayesian setting. Second, we show that in the large system limit the MMSE’s for these two settings are identical to the single measurement vector (SMV) problem with a dense measurement matrix and a block sparse signal with fixed length blocks. Third, we derive the MMSE for SMV complex CS problems by noticing that SMV complex CS is essentially an MMV problem. Fourth, we identify several performance regions for 