I'm currently working with the continuity result by Kretschmann-Schlingemann-Werner (arXiv version) for Stinespring isometries (more precisely, the following corollary to their result, cf. Appendix C in this paper):

Given channels $\Phi_1,\Phi_2\in\mathcal L(\mathbb C^{d\times d})$ with respective Kraus rank $r_1,r_2$, as well as any $m\in\mathbb N$ with $m\geq2\max\{r_1,r_2\}$, for all Stinespring isometries $V_1,V_2:\mathbb C^d\to\mathbb C^d\otimes\mathbb C^m$ of $\Phi_1,\Phi_2$ there exists $U \in\mathsf U(m)$ such that $$ \|V_1-({\bf 1}\otimes U)V_2\|_\infty\leq \sqrt{\|\Phi_1-\Phi_2\|_\diamond}\,. $$

I was interested in how good this bound is in general, which is why I got interested in finding examples of channels where $\|\Phi_1-\Phi_2\|_\diamond$ is known. However, the only literature I know of in this direction are:

• the paper of Johnston et al. (arXiv version) on the (diamond norm-)distance between unitary channels.
• the paper of Nechita et al. (arXiv version) on the distance between random channels. Therein they give an upper bound on the diamond norm in terms of the partial trace of the absolute value of the Choi-Jamiołkowski matrix, and they even characterize when this bound is saturated. Moreover, they study the distance between random channels and the maximally depolarizing channel $X\mapsto{\rm tr}(X)d^{-1}{\bf 1}_d$—as well as distances involving random unitary channels—in the asymptotic limit $d\to\infty$.
• more general results on bounding the diamond norm, e.g., the inequality $\|\Phi-{\rm id}\|_\diamond\leq 2\sqrt{\|\Phi-{\rm id}\|_{1\to 1}}$ between diamond norm and the operator norm induced by the trace norm from Watrous's book.

From these, the only analytic result on channel distances seems to be Theorem 12 in the Johnston paper which states that $\|{\rm id}-U(\cdot)U^*\|_\diamond$ is equal to "the diameter of the smallest closed disc that contains all of the eigenvalues of $U$".

Beyond that I proved the following distance results myself (but I am not yet aware of any paper or book that features them): The dephasing channel in dimension $d$ by $\Phi_d$ (i.e. $\Phi_d$ sets all off-diagonal elements of the input to zero) satisfies $$\|{\rm id}-\Phi_d\|_\diamond=2-\frac2d\,.$$ Moreover, for the depolarization channel $\Phi_{\bf 1}:X\mapsto{\rm tr}(X)d^{-1}{\bf 1}$ it holds that $$\|{\rm id}-\Phi_{\bf 1}\|_\diamond=2-\frac2{d^2}\,.$$

Now my question is twofold:

1. Are these distance results regarding dephasing and depolarization known, and if so, in what paper/book can I find them?
2. Beyond what I listed above, are there any resources which contain distance results of this type (i.e. analytic expressions for $\|{\rm id}-\Phi\|_\diamond$ for specific channels $\Phi$)? For example, a generalization of the depolarization-result would be to consider arbitrary reset channels $\Phi_\rho:X\mapsto {\rm tr}(X)\rho$ (with $\rho$ some state). While numerics suggest a closed-form expression of $\|{\rm id}-\Phi_\rho\|_\diamond$, my proof of the depolarization result breaks down when replacing ${\bf 1}/d$ by an arbitrary state.

I'm currently working with the continuity result by Kretschmann-Schlingemann-Werner (arXiv version) for Stinespring isometries (more precisely, the following corollary to their result, cf. Appendix C in this paper):

Given channels $\Phi_1,\Phi_2\in\mathcal L(\mathbb C^{d\times d})$ with respective Kraus rank $r_1,r_2$, as well as any $m\in\mathbb N$ with $m\geq2\max\{r_1,r_2\}$, for all Stinespring isometries $V_1,V_2:\mathbb C^d\to\mathbb C^d\otimes\mathbb C^m$ of $\Phi_1,\Phi_2$ there exists $U \in\mathsf U(m)$ such that $$ \|V_1-({\bf 1}\otimes U)V_2\|_\infty\leq \sqrt{\|\Phi_1-\Phi_2\|_\diamond}\,. $$

I was interested in how good this bound is in general, which is why I got interested in finding examples of channels where $\|\Phi_1-\Phi_2\|_\diamond$ is known. However, the only literature I know of in this direction are:

• the paper of Johnston et al. (arXiv version) on the (diamond norm-)distance between unitary channels.
• the paper of Nechita et al. (arXiv version) on the distance between random channels. Therein they give an upper bound on the diamond norm in terms of the partial trace of the absolute value of the Choi-Jamiołkowski matrix, and they even characterize when this bound is saturated. Moreover, they study the distance between random channels and the maximally depolarizing channel $X\mapsto{\rm tr}(X)d^{-1}{\bf 1}_d$—as well as distances involving random unitary channels—in the asymptotic limit $d\to\infty$.
• more general results on bounding the diamond norm, e.g., the inequality $\|\Phi-{\rm id}\|_\diamond\leq 2\sqrt{\|\Phi-{\rm id}\|_{1\to 1}}$ between diamond norm and the operator norm induced by the trace norm from Watrous's book.

From these, the only analytic result on channel distances seems to be Theorem 12 in the Johnston paper which states that $\|{\rm id}-U(\cdot)U^*\|_\diamond$ is equal to "the diameter of the smallest closed disc that contains all of the eigenvalues of $U$".

Beyond that I proved the following distance results myself (but I am not yet aware of any paper or book that features them):

Let $\Phi_d$ denote the dephasing channel (i.e. $\Phi_d$ sets all off-diagonal elements of the input to zero) and let $\Phi_{\bf 1}:X\mapsto{\rm tr}(X)d^{-1}{\bf 1}$ denote the depolarization channel in dimension $d$. Then $$\|{\rm id}-\Phi_{\bf 1}\|_{1\to 1}=\|{\rm id}-\Phi_d\|_{1\to 1}=\|{\rm id}-\Phi_d\|_\diamond=2-\frac2d\leq2-\frac2{d^2}=\|{\rm id}-\Phi_{\bf 1}\|_\diamond \,.$$

Now my question is twofold:

1. Are these distance results regarding dephasing and depolarization known, and if so, in what paper/book can I find them?
2. Beyond what I listed above, are there any resources which contain distance results of this type (i.e. analytic expressions for $\|{\rm id}-\Phi\|_\diamond$ for specific channels $\Phi$)? For example, a generalization of the depolarization-result would be to consider arbitrary reset channels $\Phi_\rho:X\mapsto {\rm tr}(X)\rho$ (with $\rho$ some state). While numerics suggest a closed-form expression of $\|{\rm id}-\Phi_\rho\|_\diamond$, my proof of the depolarization result breaks down when replacing ${\bf 1}/d$ by an arbitrary state.