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This answer won't actually give you a bound, but will provide some information that may help you in your search. You may be able to find an answer in the random matrix theory literature if you translate the question into different terms, as I will describe.
First, suppose that $\Phi_0(X) = U_0 X U_0^{\dagger}$ and $\Phi_1(X) = U_1 X U_1^{\dagger}$ are any ...
5
For arbitrary linear super-operators $U_j$ and $V_j\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\diamond}\def\le{\leqslant} $, we have
$$\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\diamond}\def\le{\leqslant} \begin{aligned}
\D(U_1 U_2, V_1 V_2)
&= \Dn{U_1 U_2 - V_1 V_2}
\\&\le \Dn{U_1 U_2 - V_1 U_2} + \Dn{V_1 U_2 - V_1 V_2}
\\&= ...
4
Intuition
The expression $\|\mathcal{A} - \mathcal{I}\|_\diamond$ quantifies how close the channel $\mathcal{A}$ is to the identity channel $\mathcal{I}$ which is the channel that preserves quantum information perfectly. In order for a channel to transfer quantum information well, it must preserve both diagonal and off-diagonal elements of the input density ...
4
Yes, in fact there exists Stinespring dilations such that
$$\frac{\|N_1-N_2\|_{cb}}{\sqrt{\|N_1\|_{cb}}+\sqrt{\|N_2\|_{cb}}}\leq \|V_1-V_2\|\leq \sqrt{\|N_1-N_2\|_{cb}}$$ where the distance between the isometries is the in terms of the operator norm and $N_1,N_2$ are unital completely positive maps and $V_1,V_2$ are their Stinespring isometries.
A unital ...
4
The result you're looking for is effectively Proposition 19 of the paper: Almost all quantum channels are equidistant; which I'm rewriting here for convenience:
Let $U, V \in \mathcal{U}(d)$ be two independent random variables, at least one of them being Haar-distributed. Then, with overwhelming probability as $d \rightarrow \infty$, the quantum channels $\...
3
Let $V_i=U_i^\dagger(A\otimes I_K)U_i$ and $\sigma=\rho\otimes |0\rangle\langle0|$, then we have that $V_i$ is unitary and
$$\|\rho_1-\rho_2\|_1=\|tr_K\left((V_1\sigma V_1^\dagger -V_2\sigma V_2^\dagger\right)\|_1\\
\leq \|V_1\sigma V_1^\dagger -V_2\sigma V_2^\dagger\|_1\\
=\|V_1\sigma V_1^\dagger -V_1\sigma V_2^\dagger+V_1\sigma V_2^\dagger-V_2\sigma V_2^\...
3
QETLAB usually deals with channels as Choi operators. You can convert your Kraus operators to the Choi matrix by providing the Kraus operators as a cell array. Example with the amplitude damping channel below.
>> damp = 0.3;
>> K = { diag([1,sqrt(1-damp)]); [0,sqrt(damp);0,0] };
>> ChoiMatrix(K)
ans =
1.0000 0 0 0....
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