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Gaussian states appear everywhere in quantum physics - from condensed matter, to quantum optics and quantum field theory in curved spacetimes. In this talk I aim for three objectives related to Gaussian states: First, I discuss their partner mode entanglement structure which is well known, but maybe not widely known. This is the fact that a pure Gaussian state under any bipartition decomposes into a sum of two-mode squeezed states, for example, in the Unruh effect or in Einstein-Bose condensates. Second, I review the powerful Kähler structure formalism for Gaussian states. It captures the geometry of Gaussian states, it is very practical for applications, and it treats both bosons and fermions simultaneously. Third, to showcase an example connecting bosonic and fermionic systems, I will present an entanglement duality which arises in supersymmetric Gaussian states [1]. Here, bosonic and fermionic partner subsystems, canonically identified by the supercharge construction, exhibit an intriguing duality of their entanglement spectra. (Time permitting, as a special application, we consider topological insulators and superconductors and their SUSY partners, discussing the recently derived classification of supercharges in this context [2].)
[1] Jonsson, Robert H., Lucas Hackl, and Krishanu Roychowdhury. “Entanglement Dualities in Supersymmetry.” Physical Review Research 3, no. 2 (June 16, 2021): 023213.
[2] Gong, Zongping, Robert H. Jonsson, and Daniel Malz. “Supersymmetric Free Fermions and Bosons: Locality, Symmetry, and Topology.” Physical Review B 105, no. 8 (February 24, 2022): 085423.