Efficient computability of the ε-close-degradable distance parameter

Ascertain whether the optimization problem \hat{ε}_Φ = inf_{Ψ,Θ} ||Φ − Ψ||_diamond subject to Ψ being degradable (i.e., there exists a channel Θ with Ψ^c = Θ ∘ Ψ) can be solved or efficiently approximated; determine its computational complexity class or provide efficient algorithms.

Background

Beyond ε-degradability, the paper defines ε-close-degradability via the diamond-norm distance to the set of degradable channels and formulates an optimization problem \hat{ε}Φ capturing this minimal distance. Unlike the εΦ parameter for ε-degradability, which is computable via semidefinite programming, \hat{ε}_Φ involves non-linear constraints and is not known to be efficiently solvable.

Clarifying whether \hat{ε}_Φ is efficiently computable (or even approximable) would significantly impact the practical use of the closeness-based notion, enabling tighter capacity bounds and facilitating numerical studies of proximity to degradability.

References

Note that unlike the optimization problem opt:epsi which can be phrased as an SDP and as a consequence can be solved efficiently, it is unclear if \hat \varepsilon_{\Phi} can be computed efficiently.

— Approximate Degradable Quantum Channels  (1412.0980 - Sutter et al., 2014) in Appendix A: "Approximate degradabiliy versus closeness to degradable channels"