If information codewords span over sufficiently large number of channel realizations, the achievable maximum information rate is the ergodic channel capacity. Alternatively, one can say that the channel exhibits fast fading (or block fading) characteristic. It was shown in [1] that the ergodic capacity of flat fading MIMO systems can be expressed as: $$C =E_\mathbf{H}\left \{ \underset{\mathbf{x}:\rm{tr}\left(\mathbf{Q}\right)\leqslant N_t} {\max}\rm{log}_{2} \rm{det}(I_{N_r}+\frac{P_T}{\sigma^2 N_t}\mathbf{H}\mathbf{Q}\mathbf{H}^{H})\right \}$$ where $\mathbf{Q}$ denotes the covariance matrix of the symbol vector $\mathbf{x}$, $P_T$ is the total transmit power and $\sigma ^2$ is the noise power.
In [2], it was proven that the optimal input covariance matrix is $\mathbf{Q} =\mathbf{I}_{N_t} $ for open-loop i.i.d. MIMO systems, i.e., Channel State Information (CSI) is only available at receiver. Thus, the ergodic capacity is simplified to $$ C =E_\mathbf{H}\left \{ \rm{log}_{2} \rm{det}(I_{N_r}+\frac{P_T}{\sigma^2 N_t}\mathbf{H}\mathbf{H}^{H})\right \} = E_\mathbf{H}\left \{ \sum_{i=1}^{\rm{min}\left(N_t ,N_r\right)}\log_{2} (1+\frac{P_T}{\sigma^2 N_t}\lambda_{i}^{2})\right \} $$ where $\lambda_{i}$ denotes the singular value of the channel matrix $\mathbf{H}$.

When CSI is also available to the transmitter, the achievable channel capacity is larger by optimizing the input covariance matrix $\mathbf{Q}$ using Water-filling algorithm, which will be discussed in the next slide.