Centered limiting law and sharper variance at the balanced cut

Determine the centered limiting law of the von Neumann entanglement entropy for the fixed nonzero equal-squeezing Haar ensemble at the balanced proportional bipartition ratio r=1/2, and sharpen the general variance bound Var S_{1,n}=O_s(log^2 n).

Background

The paper proves von Neumann weak typicality for proportional subsystem sizes in the fixed nonzero equal-squeezing Haar ensemble, but it separately establishes that strong typicality fails whenever the limiting subsystem ratio r differs from 1/2. At the balanced ratio, the limiting Jacobi spectral support reaches the endpoint corresponding to the logarithmic singularity of the entropy profile’s derivative.

Because of this endpoint singularity, the standard C1 Jacobi linear-statistics central limit theorem used to rule out strong typicality away from the balanced cut cannot be applied directly at r=1/2. The paper therefore leaves unresolved both the centered fluctuation law at the balanced cut and improvement of its general O_s(log2 n) variance estimate.

References

Determining its centered limiting law, as well as sharpening the general O(\log2 n) variance bound, remains open.

— Weak Typicality of von Neumann Entanglement Entropy in Gaussian Boson Sampling  (2608.17274 - Zhao, 18 Aug 2026) in Section 6, subsection “Limitations and open directions”