Quantum Sliced Wasserstein Embedding: Geometry and Learning from Homodyne Data

Continuous-variable quantum optics underlies photonic computing, bosonic error correction and precision sensing. Information is obtained by homodyne detection of the quadratures of an optical mode, and each shot produces one sample from a quadrature distribution and destroys the state. Extracting what is needed from few shots is the central practical problem. We bring sliced optimal transport, developed in machine learning for exactly this kind of data, to the homodyne record, and compare states through their measured slices with no reconstruction. Our instrument is the quantum sliced Wasserstein embedding, a map from a homodyne record to a Euclidean vector whose distances are the sliced Wasserstein distance between quadrature marginals. That relationship gives the embedding a metric structure in which displacement and squeezing of the source are locally orthogonal. We confirm the geometry on a grid of displaced and squeezed states, whose multidimensional scaling image reproduces the displacement lattice at physical spacing and stretches along squeezing as the metric predicts. We then apply the embedding to cat states, a resource widely used in quantum computing, and regress their displacement and squeezing from shot records, comparing the error to the Cramer-Rao bound. Moment, energy-distance and kernel mean-embedding features of the same records serve as baselines. Under a linear map fitted on a few dozen labeled states the embedding comes within a factor two of the bound on both parameters and is the best feature at every shot budget. The kernel features match it only with a nonlinear regressor and ten times the training states. The embedding is therefore the representation to use when a Gaussian operation on a non-Gaussian state must be read from few shots and few labels.