Determine whether the distinct SRE scaling regimes are coincidental

Determine whether the apparent distinction between the $t<1$ and $t>1$ regimes produced by the stabilizer Rényi entropy in the deformed Dyck-Fredkin spin chain is a mere coincidence, despite the entanglement entropy exhibiting constant scaling in both regimes.

Background

The paper compares the stabilizer Rényi entropy (SRE), entanglement entropy, and spectral gap of the deformed Dyck-Fredkin spin chain. For deformation parameter $t<1$, the SRE is numerically found to scale extensively as $\Theta(N)$, whereas for $t>1$ it scales as $\Theta(1)$. In contrast, the entanglement entropy has $\Theta(1)$ scaling on both sides of the undeformed point $t=1$, even though the spectral gap behaves qualitatively differently in the two regimes.

The unresolved issue is whether the SRE's ability to distinguish the $t<1$ and $t>1$ regimes reflects meaningful information about the underlying quantum many-body system or is merely accidental. Resolving this could clarify whether non-stabilizerness captures physical properties that are not encoded by entanglement entropy.