Projective measurements on quantum many-body systems produce snapshots — binary strings sampled from the Born distribution. The standard approach: learn the quantum state from these snapshots, then compute distances between states. This requires state tomography or representation learning, which scales poorly and demands prior knowledge of the relevant order parameters.
The paper inverts the pipeline. Instead of learning states and then computing distances, it learns distances directly from snapshots. A neural discriminator estimates Csiszár f-divergences between the probability distributions underlying pairs of snapshot collections. No state reconstruction, no order parameter selection — just pairwise distances between measurement distributions.
The distances are sufficient. Phase boundaries appear as ridges in the distance matrix. Critical exponents emerge by connecting the inferred divergences to the Fisher information metric. Universality classes are identified by comparing distance scaling across different systems. The method recovers known results across Ising models, toric codes, and fermionic systems — from conventional ordered phases to topological order to higher-order correlations.
The structural insight: phase diagrams are distance structures, not state structures. You don't need to know what the states are to know how different they are. The phase boundary is the locus where neighboring parameter values produce maximally distinguishable measurement distributions. This is a geometric fact about the space of probability distributions, not a physical fact about the quantum states — the physics enters only through the measurement, and the geometry does the rest. Clustering in distance space is phase identification without phase characterization.