Definition of percolation for oscillatory asymptotic entanglement

Determine how entanglement percolation should be rigorously defined for series quantum networks when the sponge-crossing entanglement has subsequences converging to positive values and other subsequences converging to zero.

Background

The paper defines percolation using a positive limit inferior of the sponge-crossing entanglement and failure using a zero limit superior. These definitions do not classify sequences whose limit inferior is zero while their limit superior is positive.

Such behavior can occur when the asymptotic entanglement oscillates or depends irregularly on network scale. The paper explicitly identifies the need for a rigorous classification of this intermediate case.

References

It should be noted that our definition of entanglement percolation for series QNs in Sec.~\ref{sec-series} remains incomplete. For instance, when both $\overline\lim_{n\to \infty}E_{\rm SC}{(n)}>0$ and $\varliminf_{n\to \infty}E_{\rm SC}{(n)}=0$ hold--certain subsequences converge to positive values while others converge to zero--how should such percolation behavior be rigorously defined?

— An Operator-Theoretic Model of Entanglement Percolation in Series-Parallel Quantum Networks  (2609.26367 - He et al., 22 Sep 2026) in Conclusion and discussion, item 5; see also Section 6.1