Computability and finite-dimensional characterization of κ

Determine whether the channel quantity \(\kappa(\Phi)\) is computable, whether finite-dimensional witnesses or finite auxiliary-dimension bounds can characterize it, and whether \(\kappa\) is continuous as a function of the finite-dimensional quantum channel \(\Phi\).

Background

The paper characterizes the optimal closed-device dimension and initial-entropy rates using κ(Φ)\kappa(\Phi), defined through an ordered limit of an infimum over finite extensions with unrestricted auxiliary dimension. Although the main rate region is established, the construction does not provide an algorithm for evaluating κ\kappa, a finite bound on the required witness dimension, or a continuity theorem with respect to the channel.

The authors explicitly identify these computational and structural issues as unresolved. The lower bound in Proposition~\ref{prop:kappa-lower} is readily evaluable, but Proposition~\ref{prop:interior} demonstrates that the available lower and upper bounds can remain separated.

References

Three questions are left open here. Computability: $\kappa$ is an ordered limit of an infimum over extensions of unbounded dimension, and no algorithm, finite witness bound or continuity in $\Phi$ is asserted; Proposition~\ref{prop:kappa-lower} gives only a lower bound that is easy to evaluate, and Proposition~\ref{prop:interior} leaves a gap of a factor of about three between that bound and the exhibited upper bound.

— Bath dimension and initial entropy for closed repeated use of a quantum channel  (2609.18267 - Douglas, 16 Sep 2026) in Section 6, “Conclusion and open questions” (\label{sec:conclusion})