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We address the problem of the reconstruction of quantum covariance matricesusing the notion of Lagrangian and symplectic polar duality introduced inprevious work. We apply our constructions to Gaussian quantum states whichleads to a non-trivial generalization of Pauli's reconstruction problem and westate a simple tomographic characterization of such states.

Polar Duality and the Reconstruction of Quantum Covariance Matrices from Partial Data