Unravelling the Li-Haldane Conjecture with the Projected Ensemble

Abstract: The entanglement spectra of fractional quantum Hall states contain universal fingerprints of their underlying topological order, as posited by the Li-Haldane conjecture. In this work, we uncover a finer universal structure within the entanglement spectra unravelled by projective measurements. Concretely, we study the projected ensemble of fractional quantum Hall states, defined as the collection of quantum states on a subsystem conditioned on measurement outcomes of its complement. We find that this ensemble exhibits a hidden hierarchy inside the Li-Haldane edge manifold: by conditioning on measurement outcomes, the entanglement spectrum's support is split into measurement-dependent sectors whose ranks we demonstrate are fixed by conformal field theory counting, an observation we dub the measurement-resolved Li-Haldane conjecture. For the non-Abelian Moore-Read state, this hierarchy is particularly rich: each parity-resolved edge manifold contains internal subspaces whose dimensions reproduce the conformal field theory counting of the opposite-parity sector. This structure persists even in realistic Coulomb-interacting ground states, establishing the projected ensemble as a sharp new probe of topological order beyond what the entanglement spectrum alone can detect.