product of two positive semi definite matrices $\mathbf{P}$, $\mathbf{Q}$. Using the fact, that $\mathbf{\tilde{H_e}}\mathbf{\tilde{H_e}}^H$ has the same ordered eigenvalues as $\mathbf{\tilde{H}}\mathbf{\tilde{H}}^H$ and the inequality $\mathrm{tr}(\mathbf{PQ})\le\sum_j
\mu_k(\mathbf{P})\mu_k(\mathbf{Q})$ [@MarshallOlkin] yields $\sum\limits_{k=1}^{L_1}\mu_{k}^2\le L_1$ and we arrive at the final bound $$\begin{aligned}
\nonumber
\Delta \leq \frac{1}{\ln(2)}\mathbb{E}\left[
  \sum\limits_{k=1}^{\nicefrac{L_1}{2}}\mu_{k}^2\right]
  \le \frac{L_1}{2 \ln(2) }.\end{aligned}$$

In addition to that, we have the loss between $C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)$ and $C\left(\rho,n_T,n_R\right)$. Approximating  and  for high SNR as $$\begin{aligned}
& C_{\mathrm{Jen}} \left(\rho,n_T,n_R\right)= \log_2\left(1+ \sum_{i=1}^L
\binom{L}{i}\frac{K!}{(K-i)!}\left(\frac{\rho}{n_T}\right)^i\right) \nonumber\\
 \approx & \log_2\left(1+ \frac{K!}{(K-L)!}
\left(\frac{\rho}{n_T}\right)^L\right) \nonumber\\
 = & \log_2\left( \frac{(K-L)!}{K!}+ \left(\frac{\rho}{n_T}\right)^L\right)+\log_2\left(\frac{K!}{(K-L)!} \right) \nonumber\\
 \approx  & \log_2\left( 1+ \left(\frac{\rho}{n_T}\right)^L\right) \approx   \log_2\left( \left( 1+
\frac{\rho}{n_T}\right)^L\right) \nonumber \\
= & L\log_2\left(1+\frac{\rho}{n_T}\right)\label{eq:ApproxCapNTNR}\end{aligned}$$ and $$\begin{aligned}
& C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right) \nonumber \\
 \overset{(a)}{\geq} &
\frac{n_T}{2} \log_2\left(1+\frac{\rho}{n_T}\exp\left(\frac{2}{n_T} \sum_{j=1}^{\frac{n_T}{2}}\sum_{p=1}^{2n_R-j}\frac{1}{p}-\gamma\right)\right) \nonumber\\
 = & \frac{n_T}{2} \log_2\left(\exp\left(-\frac{2}{n_T}
\sum_{j=1}^{\frac{n_T}{2}}\sum_{p=1}^{2n_R-j}\frac{1}{p}+\gamma\right)+\frac{\rho}{n_T}\right) \nonumber \\
& + \frac{n_T}{2\ln(2)} \left(\frac{2}{n_T}
\sum_{j=1}^{\frac{n_T}{2}}\sum_{p=1}^{2n_R-j}\frac{1}{p}-\gamma\right) \nonumber \\
\approx & \left(\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right),\label{eq:ApproxCapNT22NR}\end{aligned}$$ where $(a)$ follows from applying Jensen’s inequality to . With  and , the loss between $C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)$ and $C\left(\rho,n_T,n_R\right)$ is quite accurately described by $$\begin{aligned}
\nonumber
& C\left(\rho,n_T,n_R\right)-C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right) \\
& \approx
 \left(L-\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right), \quad[\text{case }n_T < 2n_R] \nonumber.\end{aligned}$$ Finally, the absolute loss for $n_T < 2n_R$ between the ergodic capacity of a MIMO system and the stacked scheme is given by $$\begin{aligned}
\nonumber
& \left(L-\frac{n_T}{2}\right)\log_2\left(1+\frac{\rho}{n_T}\right) \leq C\left(\rho,n_T,n_R\right)-R_{sA} \\
& \leq \frac{n_T}{2 \ln(2) 