Existence of continuous Stinespring isometry curves tracking diamond-norm dynamics

Investigate whether, given time-dependent quantum channels Φ1(t), Φ2(t), there exist a natural number m and sufficiently regular—at least locally absolutely continuous—curves of Stinespring isometries V1(t), V2(t): C^n → C^k ⊗ C^m of Φ1(t) and Φ2(t), respectively, such that ||V1(t) − V2(t)||_∞ ≤ √(2 ||Φ1(t) − Φ2(t)||_⋄) holds for all times t.

Background

The authors recently studied dynamic Stinespring representations and showed approximate existence of such curves. They now ask for stronger existence and continuity properties that would ensure a pointwise inequality matching the diamond-norm distance between time-dependent channels. This question is not implied by the static conjecture due to the added regularity constraints.

References

Let us conclude by presenting two related open questions:

Given dynamic processes \mathsf\Phi_1,\mathsf\Phi_2 do there exist m\in\mathbb N as well as sufficiently regular — but at least locally absolutely continuous — curves of Stinespring isometries \mathsf V_1,\mathsf V_2 such that |\mathsf V_1(t)-\mathsf V_2(t)|{\infty}\leq\sqrt{2|\mathsf\Phi_1(t)-\mathsf\Phi_2(t)|{\diamond}} for all t? Be aware that~eq:open_2 would not be a direct consequence of Conjecture 1 due to the additional continuity requirement on \mathsf V_1,\mathsf V_2.

eq:open_2:

∥V1(t)−V2(t)∥∞≤2∥Φ1(t)−Φ2(t)∥⋄\|\mathsf V_1(t)-\mathsf V_2(t)\|_{\infty}\leq\sqrt{2\|\mathsf\Phi_1(t)-\mathsf\Phi_2(t)\|_{\diamond}}

— Progress on the Kretschmann-Schlingemann-Werner Conjecture  (2308.15389 - Ende, 2023) in Section 4, Conclusions and Outlook (Open Questions)