Photograph the quantum state through its covariance matrix, not its density matrix.
Multimode Gaussian state tomography (arXiv:2603.21380): reconstruct the full quantum state of an optical system by estimating its covariance matrix directly, using maximum-likelihood estimation. Traditional quantum state tomography reconstructs the density matrix — an object whose dimension grows exponentially with the number of modes. For Gaussian states (the states relevant to most continuous-variable quantum computing), the density matrix is redundant: all information is in the covariance matrix, which grows only quadratically.
Demonstrated on six-mode graph states with different topologies, a six-mode GHZ state, and a ten-mode fully connected graph state. The reconstructed covariance matrices yield fidelities, entanglement detection, squeezing levels, and noise characterization.
The method ensures physical covariance matrices — the maximum-likelihood estimate is constrained to satisfy the uncertainty principle. Conventional tomography can produce unphysical estimates (matrices that violate quantum mechanics) due to statistical noise. The constraint eliminates this artifact.
The structural insight: the right representation makes the impossible tractable. Full density-matrix tomography of a ten-mode state is computationally prohibitive. Covariance-matrix tomography of the same state is practical — same physics, different representation, different scaling. The Gaussian assumption isn't a limitation for these states — it's what makes them measurable. The constraint (Gaussianity) is a gift, not a restriction, because it reduces the representation from exponential to polynomial. Knowing what your state is determines how efficiently you can measure it.