Diamond norm distance bound on Stinespring dilations of channels

Question

The diamond distance between two channels $\Phi_0$ and $\Phi_1$ is defined in this answer.

$$ \| \Phi_0 - \Phi_1 \|_{\diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 \otimes \operatorname{Id}_k)(\rho) \|_1 $$ where $\operatorname{Id}_k$ denotes the identity channel from $M_k(\mathbb{C})$ (the set of $k\times k$ complex matrices) to itself, $\| \cdot \|_1$ denotes the trace norm, and the supremum is taken over all $k \geq 1$ and all density matrices $\rho$ chosen from $M_{nk}(\mathbb{C}) = M_n(\mathbb{C}) \otimes M_{k}(\mathbb{C})$.

Let $N_1$ and $N_2$ be two completely positive trace nonincreasing maps that satisfy

$$\|N_1 - N_2\|_\diamond\leq \varepsilon.$$

For any channel $N_{A\rightarrow B}$, we define its Stinespring dilation to be an isometry $V_{A\rightarrow BE}$ such that $\text{Tr}_E(V\rho V^\dagger) = N(\rho)$.

Can one show that there exist Stinespring dilations $V_1$ and $V_2$ of $N_1$ and $N_2$ respectively such that we also have a bound on

$$\|V_1 - V_2\|_\diamond$$

in terms of $\varepsilon$?

Answer 1

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 completely positive map $N_i(X)=V_i^*\pi(X)V_i$ is dual to a CPTP map, so that is something to be aware of here...

See https://arxiv.org/pdf/0710.2495.pdf

Note: above the $cb$-norm is the completely bounded "operator" norm, where as the completely bounded trace norm (or diamond norm) is more common in quantum information. I believe that this does bound the diamond norm since, at least for finite dimensional maps (completely postive maps between matrix algebras) we have $$\|V_1-V_2\|_\diamond =\sup_{dim(H)}\|(V_1-V_2)\otimes id_{B(H)}\|_1\\\leq d\cdot \sup_{dim(H)}\|(V_1-V_2)\otimes id_{B(H)}\|_\infty=d\|V_1-V_2\|_{cb}\leq d\sqrt{\|N_1-N_2\|_{cb}},$$ where $d$ is the dimension of the space on which $(V_1-V_2)\otimes id_{B(H)}$ acts, and $B(H)$ is the set of bounded operators on $H$.