In light of the question given in why only extrinsic information is passed in turbo decoding/equalization, why not a posteriori information? if we have different observation provided to two linear minimum mean squared error (LMMSE) equalizers, the extrinsic information provided from the first equalizer to the second one would be the same as the a posteriori estimate of the first equalizer since we have different observations.
In detail, if we have two observations, $\bf r_1$ and $\bf r_2$, where $\bf{r}_1=Hx+w_1$, $\bf{r}_2=Hx+w_2$, $\bf H$ is a $NXN$, and $\bf w_1, w_2$ are Gaussian noise vectors of length $N$, and we have two equalizers A and B performing LMMSE on $\bf r_1, r_2$ respectively, then the extrinsic information passed to equalizer B from A would be the LMMSE estimate of obtained in A.
If the above is true, how can turbo technique improves the performance?