Ergodic MIMO Mutual Information: Twenty Years After Emre Telatar

In the celebrated work of Emre Telatar in the year 1999 (14274 citations to date), it was shown that the expected value of the mutual information \begin{equation*} $\mathrm{I}$ = \ln\det\left( $\mathbf{I}$m + \frac{1}{t} $\mathbf{H}\mathbf{H}$^{\dagger} \right) \end{equation*} of an $m\times n$ MIMO Rayleigh channel matrix $\mathbf{H}$ with a SNR $1\mathrm{/}t$ can be represented as an integral involving Laguerre polynomials. We show, in this work, that Telatar’s integral representation can be explicitly evaluated to a finite sum of the form \begin{equation*} $\mathbb{E}$!\left[$\mathrm{I}$\right]=\sum{k=0}^{n+m-3}a_{k}t^{k}+\rm e^{t}~$\text{Ei}$(-t)\sum_{k=0}^{n+m-2}b_{k}t^{k}, \end{equation*} where $\text{Ei}(-t)$ is the exponential integral and $a_k$, $b_k$ are known constants that do not dependent on $t$. The renewed interest in this classical information theory problem came from, quite surprisingly, the recent development in quantum information theory.