Phase transitions can generate intricate spatial organization invisible to spatially averaged probes, yet quantitative measures of this organization remain limited. Here we develop "entropy imaging," an information-theoretic technique that harnesses lossless compression to quantify spatial complexity, predictability, and order in quantum materials. Using this approach, emergent spatial textures can be imaged and converted into an experimentally-accessible observable: the computable information density (CID), a compression-based measure of information complexity that approaches the Shannon entropy rate in the large-system limit. Applied to millikelvin microwave impedance microscopy of graphene, the method reveals a pronounced maximum in the CID near quantum Hall transitions, a quantitative measure of the information-rich spatial organization of the critical landscape. A disordered Hofstadter model reproduces this behavior and shows that the CID can identify the quantum critical point from a sequence of images, without ensemble averaging over disorder realizations. The same framework tracks fractional quantum Hall transitions and resolves isolated fractional puddles that coexist with a disordered background. By contrast, the trivial band-edge crossover in a MoSe2/WS2 moiré heterobilayer exhibits qualitatively different behavior, suggesting that the pronounced CID enhancement observed at the quantum Hall critical point is not simply a generic feature of a conductive-insulating crossover. We further show that the geometric information encoded in CID is not reducible to a correlation length, and it can even distinguish spatial textures with identical multifractal statistics. Together, these results establish lossless compression as a framework for quantifying emergent spatial organization in quantum materials, with portability across imaging modalities and physical systems.
Phase transitions can generate intricate spatial organization invisible to spatially averaged probes, yet quantitative measures of this organization remain limited. Here we develop "entropy imaging," an information-theoretic technique that harnesses lossless compression to quantify spatial complexity, predictability, and order in quantum materials. Using this approach, emergent spatial textures can be imaged and converted into an experimentally-accessible observable: the computable information density (CID), a compression-based measure of information complexity that approaches the Shannon entropy rate in the large-system limit. Applied to millikelvin microwave impedance microscopy of graphene, the method reveals a pronounced maximum in the CID near quantum Hall transitions, a quantitative measure of the information-rich spatial organization of the critical landscape. A disordered Hofstadter model reproduces this behavior and shows that the CID can identify the quantum critical point from a sequence of images, without ensemble averaging over disorder realizations. The same framework tracks fractional quantum Hall transitions and resolves isolated fractional puddles that coexist with a disordered background. By contrast, the trivial band-edge crossover in a MoSe2/WS2 moiré heterobilayer exhibits qualitatively different behavior, suggesting that the pronounced CID enhancement observed at the quantum Hall critical point is not simply a generic feature of a conductive-insulating crossover. We further show that the geometric information encoded in CID is not reducible to a correlation length, and it can even distinguish spatial textures with identical multifractal statistics. Together, these results establish lossless compression as a framework for quantifying emergent spatial organization in quantum materials, with portability across imaging modalities and physical systems.
Wu et al. (arXiv:2610.35790) introduced entropy imaging for quantum matter: a microscopy image is serialized along a Hilbert curve, compressed with a lossless algorithm, and the computable information density (CID) is used as a measure of spatial complexity. Applied to microwave impedance microscopy of graphene, CID peaks near quantum Hall transitions, tracks fractional transitions, and resolves fractional puddles in a local map, while a trivial band-edge crossover does not show the same enhancement. This preprint offers an information-dynamics reading of that result. In this reading, a microscopy image is a projection, and CID is treated as an indicator of the arrangement information in the projection. The reading changes no number in Wu et al. No new data are reported. All numbers are Wu et al.'s as reported. Three falsifiable predictions are stated, testing the CID diagnostic.
Coarse-grained measurements offer a scalable alternative to full state tomography for characterizing complex quantum dynamics. We show that observational entropy (OE), an information-theoretic entropy defined directly from finite-resolution measurement outcomes, provides a unified and experimentally accessible framework for quantifying chaos and probing criticality. From probing the insulator-metal crossover in the Aubry-Andre model to tracking the gradual destruction of Kolmogorov-Arnold-Moser tori in the Kicked Rotor, derivatives of OE provide an accurate and unified diagnostic of probing these transitions. In both cases, the critical points extracted from dynamical evolution and eigenstate analyses converge to the exact theoretical values once the observational resolution exceeds a finite threshold. In the chaotic limit, OE exhibits a linear behavior within the Ehrenfest time regime, and its slope defines an observable Lyapunov exponent. Using a Pretty Good Measurement correction to the Husimi phase-space distribution, this entropy-production rate quantitatively reproduces the classical Lyapunov exponent in both the standard and singular kicked rotors. Our results establish OE as a compact information-theoretic bridge between classical instability, quantum criticality, and realistic finite-resolution measurements.
We present a detailed study of the ground-state entanglement in disordered fractional quantum Hall liquids. We consider electrons at various filling fractions f in the lowest Landau level, with Coulomb interactions. At f=1/3,1/5, and 2/5 where an incompressible ground-state manifold exists at zero disorder, we observe a pronounced minimum in the derivative of entanglement entropy with respect to disorder. At each filling, the position of this minimum is stable against increasing system size, but its magnitude grows monotonically and appears to diverge in the thermodynamic limit. We consider this behavior of the entropy derivative as a compelling signal of the expected disorder-driven phase transition from a topological fractional quantum Hall phase to an insulating phase. On the contrary, at f=1/2 where a compressible composite fermion sea is present at zero disorder, the entropy derivative exhibits much greater, almost chaotic, finite-size effects, without a clear phase transition signal for system sizes within our exact diagonalization limit. However, the dependence of entanglement entropy with system size changes with increasing disorder, consistent with the expectation of a phase transition from a composite fermion sea to an insulator. Finally, we consider f=1/7 where compressible Wigner crystals are quite competitive at zero disorder, and analyze the level statistics of entanglement spectrum at f=1/3.
We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of electroencephalogram or functional magnetic resonance imaging measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson's correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, Z_{2}, mean field superfluid to Mott insulator, and a Berzinskii-Kosterlitz-Thouless crossover.
We investigate the role of quantum state texture in dynamical quantum phase transitions by establishing a direct connection between critical nonequilibrium dynamics and the recently introduced notion of rugosity, a measure of quantum state texture. Considering a generic quench protocol, we analyze both standard formulations of dynamical quantum phase transitions. For type-I transitions, characterized through the long-time behavior of a dynamical order parameter, we show that the time-averaged rugosity, evaluated in the eigenbasis of the pre-quench Hamiltonian, exhibits an order-parameter-like behavior across the transition. In the Lipkin-Meshkov-Glick model, this behavior is traced back to the underlying semiclassical structure, where crossing the excited-state quantum phase transition separatrix controls the redistribution of the quantum state over the pre-quench energy basis. We further demonstrate that the response function associated with the time-averaged rugosity develops a pronounced critical peak whose maximum grows algebraically with the system size, providing quantitative finite-size evidence of its sensitivity to the dynamical critical point. For type-II transitions, characterized by nonanalyticities in the Loschmidt rate function, we establish an exact connection with rugosity. For a suitable choice of basis, the rate function is exactly given by the density of rugosity, establishing a model-independent equivalence. Moreover, we show that even in physically motivated bases, such as the pre-quench energy eigenbasis, rugosity provides clear signatures of dynamical criticality. Our results establish rugosity as a sensitive probe of dynamical quantum phase transitions while highlighting its role as a basis-dependent measure of quantum state texture, thereby providing an information-theoretic perspective on nonequilibrium quantum critical phenomena.
The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.