I'm trying to get an upper bound on $R^2$ defined as $$x\sim \text{Normal}(0, \Sigma)$$ $$R^2=\frac{\|Ex'x xx'\|_2}{\|\Sigma\|_2}$$
$E$ is expectation, $\|\cdot\|$ is matrix spectral norm and $x$ is a column vector
In one dimension, the bound $R^2\le 3\text{tr}(\Sigma)$ is tight, how can I get a bound for many dimensions?
Edit empirically the following seems to hold $$R^2=\text{tr}(\Sigma)+2\|\Sigma\|_2$$