Exact Mapping from Singular-Value Spectrum of Fractal Images to Entanglement Spectrum of One-Dimensional Quantum Systems

We examine the snapshot entropy of fractal images defined by their singular values. Remarkably, the singular values for a large class of fractals are in exact correspondence with the entanglement spectrum of free fermions in one dimension. These fermions allow for an interpretation of the logarithmic scaling of the snapshot entropy, which is in agreement with the logarithmic violation formula of the area law scaling. However, the coarse-grained entropy exhibits a linear scaling owing to the degeneracy of the spectrum, in contrast with the logarithmic scaling behavior in onedimensional quantum near-critical systems.