Characterizing electronic band structures requires precise knowledge of wave functions and their quantum geometry. Here, we introduce a tomography method to reconstruct the density matrix of electron states from quasiparticle interference maps around single impurities. We consider two-orbital models on a honeycomb lattice, relevant to graphene heterostructures and direct-gap semiconductors. For on-site impurities, backscattering between time-reversed states directly maps the density matrix populations and coherences into distinct orbital contributions in the interference map. While local probes usually lack orbital resolution, these orbital contributions transform under distinct symmetry group representations and can thus be disentangled to reveal the density matrix and quantum geometric tensor of the scattering states. This establishes impurities as tomographic probes for band structures in scanning tunneling microscopy using conventional, unpolarized tips.

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• Density of states
• Electronic structure
• Geometric & topological phases
• Local density of states
• Mesoscopics
• Symmetry protected topological states
• 2-dimensional systems
• Honeycomb lattice
• Semiconductors
• Surfaces
• Thin films
• Two-dimensional electron system
• Bloch wave theory
• Green's function methods
• Scanning tunneling microscopy
• Scanning tunneling spectroscopy
• Single impurity model