Second-order finite-blocklength behavior near the unconstrained quantum capacity

Determine the optimal finite-blocklength fidelity behavior for entanglement-generation codes over the unconstrained pure-loss bosonic channel when the rate gap \(\delta_n=\log_2 M_n-nq_\eta\) is of order \(\sqrt n\), including matching achievable bounds, the dispersion coefficient, and a sharp second-order expansion.

Background

The paper proves an upper bound that forces fidelity to vanish when the positive rate gap satisfies δn/n→∞\delta_n/\sqrt n\to\infty. This identifies a regime beyond the expected second-order scale but does not characterize performance when the gap is proportional to n\sqrt n.

The unresolved problem is to determine the optimal behavior at this scale, establish corresponding achievable coding bounds, and ascertain whether the constant in the uniform variance estimate can be improved. Such results would clarify the finite-blocklength transition around the unconstrained quantum capacity of the pure-loss bosonic channel.

References

It remains to determine the optimal behavior when $\delta_n$ is of order $\sqrt n$, to obtain matching achievable bounds, and to assess whether the constant in the uniform variance estimate can be improved. The present argument neither identifies a dispersion coefficient nor proves a sharp second-order expansion.

— Strong converse for the quantum capacity of the pure-loss bosonic channel  (2609.16608 - Wilde, 15 Sep 2026) in Section 6, Conclusion and future research, paragraph “Finite-blocklength behavior near capacity”