Thermodynamic saturation of the measurement-resolved Li–Haldane bound

Prove that the conditional CFT rank bound for the conditional reduced density matrices of fractional quantum Hall states becomes saturated with probability approaching unity in the thermodynamic limit.

Background

The paper introduces a measurement-resolved Li–Haldane rank bound for conditional reduced density matrices obtained after measuring a subset of orbitals. Finite-size numerics indicate that the bound is robust and that the probability of saturation increases with system size, while exceptional nonsaturating measurement outcomes are pushed toward increasingly nonlocal patterns.

The authors explicitly state that the available finite-size calculations do not prove the asymptotic claim. Establishing probability-one saturation in the thermodynamic limit would therefore turn the observed numerical tendency into a rigorous result.

References

The numerical data show a clear tendency toward increasing saturation probability with system size. In particular, for a fixed measurement depth $m_{\mathrm B_1}$, an increasing fraction of the projected-ensemble weight is carried by outcomes that saturate Eq.~(\ref{rankcollapseconjecture}). This trend suggests that saturation becomes increasingly generic with increasing particle number and is consistent with $p_{\mathrm{sat}\rightarrow1$ in the thermodynamic limit. The finite-size violations themselves also exhibit a characteristic structure. A common non-saturating outcome consists of a contiguous string of empty orbitals, $\mathbf n_{\mathrm B_1}={00\cdots0}$, adjacent to the orbital cut. Numerically, the length of the exceptional zero string required to observe non-saturation grows approximately linearly with system size. Thus, although such exceptional outcomes persist at the finite sizes accessible to our numerics, they are displaced progressively farther from the cut as the system grows. This provides an additional indication that finite-size violations of saturation are pushed to increasingly nonlocal measurement patterns in the thermodynamic limit.

— Unravelling the Li-Haldane Conjecture with the Projected Ensemble  (2609.03833 - Spasic-Mlacak et al., 3 Sep 2026) in Appendix, Section “Saturation of Rank Collapse”

As the Li--Haldane counting depends only on these quantum numbers, it is natural to conjecture that the rank of the CRDM is determined entirely by the corresponding effective edge sector.

— Unravelling the Li-Haldane Conjecture with the Projected Ensemble  (2609.03833 - Spasic-Mlacak et al., 3 Sep 2026) in Section “Measurement-resolved Li–Haldane conjecture” and Appendix, Section “Derivation of Measurement-Resolved Conjecture”