Infinite-dimensional extension of the updated KSW inequality

Determine whether, for arbitrary complex Hilbert spaces H, K, Z and arbitrary isometries V1, V2: H → K ⊗ Z, the inequality min_{U ∈ U(Z)} ||V1 − (I_K ⊗ U)V2||_∞ ≤ √(2 || tr_Z(V1(·)V1*) − tr_Z(V2(·)V2*) ||_⋄) holds.

Background

Because the conjectured bound is independent of finite-dimensional sizes, it is natural to ask whether it extends to general (possibly infinite-dimensional) Hilbert spaces. An affirmative resolution would generalize the finite-dimensional KSW-type continuity relation between Stinespring isometries and the diamond-norm distance of the induced channels.

References

Let us conclude by presenting two related open questions:

As the bound in Conjecture 1 is independent of the dimension of the underlying spaces one may even extend the former to infinite dimensions: Given complex Hilbert spaces \mathcal H,\mathcal K,\mathcal Z and arbitrary isometries V_1,V_2:\mathcal H\to\mathcal K\otimes\mathcal Z does it hold that \min_{U\in\mathsf U(\mathcal Z)}|V_1-(\mathbbm1_{\mathcal K}\otimes U)V_2|\infty\leq \sqrt{2|{\rm tr}{\mathcal Z}(V_1(\cdot)V_1*)-{\rm tr}{\mathcal Z}(V_2(\cdot)V_2*)|\diamond}\,?

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