Orbit dimensions in linear and Gaussian quantum optics

Abstract

We study the dimension of the manifold of quantum states (called orbit) that a given quantum state of light can reach under the dynamics of linear or Gaussian quantum optics. That is, we investigate how many directions in the Hilbert space a given state can explore under these sub-universal regimes. We find that these orbit dimensions reveal fundamental insights into the structure of attainable state spaces (e.g. boson bunching does not increase the number of accessible directions) with multi-faceted consequences. By showcasing a simple way to compute this topological quantity, we reveal how it can alone yield no-go results for some transformations. Our framework is proven to hold in both discrete and continuous-variable settings, and can be used with Fock as well as phase-space representations such as the Wigner or stellar representations. We study genericity and robustness properties of orbit dimensions, and propose strategies to probe them using homodyne/heterodyne measurements on pure states, or photon counters on two copies of general states. We also highlight how under Gaussian unitaries, orbit dimensions witness non-Gaussianity of quantum states. Lastly, we rigorously establish implications of orbit dimensions for the number of directions that a bosonic variational circuit can explore in state space. Our approach sheds new light on the structure of reachable states in quantum optics, which can help practitioners understand limitations and sources of expressivity or non-Gaussianity in bosonic quantum information protocols such as quantum machine learning.