Convexity of the ansatz-based six-state entropy bound

Determine whether the lower bound on the sandwiched Rényi entropy obtained using the ansatz for \(\tau_E\) in Eq. (\ref{eq:ansatz}) is convex as a function of the Werner-state parameter \(Q\).

Background

For the six-state protocol, the paper derives three lower bounds on the relevant conditional Rényi entropy after measuring a Werner state, parameterized by QQ. Convexity in QQ is important because the numerical keyrate method replaces the entropy bound by tangent-line lower bounds and then solves the resulting convex optimization problem.

The first two entropy expressions are established to be convex, but the third is obtained only as a lower bound using an ansatz for the optimizing state τE\tau_E, rather than by evaluating the entropy exactly. Consequently, the argument used to prove convexity of the first two expressions does not apply to the third bound. The authors also report that this third bound performs less well empirically, but its convexity remains unresolved.

References

On the other hand, it is less clear whether the bound we obtained from $\tilde{H}\uparrow _\alpha(B|E)$ is convex. This is because we did not compute its exact value (which would have allowed us to applyLemma~16), but rather a lower bound on it via the ansatz $\tau_E$ in~eq:ansatz. We leave the convexity of that bound for future work, as our empirical observations below indicate it performs less well than the other bounds in practice.

— Simple QKD keyrate computations from tangent-line bounds  (2609.09847 - Goh et al., 9 Sep 2026) in Appendix, Section “Rényi entropy from measuring a Werner state,” immediately before subsection “Comparison of bounds”