Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$ m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)}, $$
where $\lambda_{\max}$ denotes the largest eigenvalue.
Now let $Q: \mathbb{R}^{d \times d} \to \mathbb{R}^{d \times d}$ be a self-adjoint, completely positive linear map satisfying the following conditions:
- $\sigma(Q) \subset [0, 1]$ (the spectrum of $Q$ is contained within $[0, 1]$),
- $Q[\mathbf{I}] \preccurlyeq \mathbf{I}$ (the action of $Q$ on the identity matrix is dominated by the identity matrix in the Löwner order).
Question
What is the tightest possible upper bound on $m[Q^k - Q^{k+1}]$ for $k \in \mathbb{N}^+$?
The best bound I have derived so far is:
$$ m[Q^k - Q^{k+1}] \leq \min\left(\sqrt{d}, \frac{1}{1 - \|Q\|} m[Q^k]\right) \leq \min\left(\sqrt{d}, \frac{1}{\|Q\|^k - \|Q\|^{k+1}}\right), $$
where $\|Q\|$ is the operator norm of $Q$.
Additional Notes:
The $\sqrt{d}$ term arises from the relationship between the spectral norm and the Frobenius norm: for a $d \times d$ matrix $A$,
$$ \|A\| \leq \|A\|_F \leq \sqrt{d} \|A\|, $$
which bounds the maximal possible value of $m[H]$ based on $H[\mathbf{I}]$ and the spectrum of $H$.
Specific Interest
Can we improve this bound, particularly by further bounding $m[Q^k - Q^{k+1}]$ in terms of $m[Q^k]$ or $m[Q^{k+1}]$?
