Laughlin quasihole geometry from a single snapshot ensemble

Can complex quantum geometry, such as quantum metrics and Berry curvature, be determined entirely from a probability distribution measured in a single basis without needing to prepare nearby states or drive the system dynamically? The study investigates whether the analytic structure of lattice Laughlin states allows a single occupation snapshot ensemble to extract relational quantum properties normally requiring multiple distinct state preparations.

The author derives an analytic relation showing that the amplitude of a lattice Laughlin quasihole factorizes into a fixed background contribution and a known configuration-wise quasihole factor. By measuring the occupation probability distribution at a generic reference position and computing configuration-wise amplitude ratios, the unknown reference phases cancel out. This yields an unnormalized overlap kernel between unprepared states. The approach is validated using exact enumeration of a Nielsen-Cirac-Sierra state on an eight-site lattice with three particles.

The central result is a single-reference kernel identity that uses measurement data from one reference position to reconstruct finite-distance overlaps, Bargmann phases, Connes distances, and local quantum geometry. Specifically, the Fubini-Study metric and Berry curvature are locked together and can be extracted via second derivatives of the reconstructed kernel. Additionally, the projective span of the quasihole family is constrained to a subspace whose dimension grows at most linearly with particle number, independent of the large ambient Hilbert space. Furthermore, local occupation readout saturates the single-copy multiparameter information bound, demonstrating optimal local efficiency.

This work bridges quantum-gas microscopy capabilities with abstract quantum geometry, offering a powerful shortcut for experiments. Instead of performing complex multi-state interferometry or dynamical driving, researchers can extract rich geometric properties of many-body quantum states from existing measurement datasets, significantly reducing the experimental overhead required to study topological defects and quantum states.