Orbit dimensions in linear and Gaussian quantum optics

We study the dimension of the manifold of quantum states (called orbit) that a given bosonic state can reach under linear or quadratic Hamiltonian evolutions. That is, we investigate how many directions in the Hilbert space a state can explore in these sub-universal regimes. After showcasing a simple way to compute orbit dimensions, we find that these topological quantities reveal fundamental insights into the structure of attainable state spaces (e.g., boson bunching does not increase the number of accessible directions) with multifaceted consequences. First, we illustrate how they can alone yield no-go results for some transformations. We then propose ways to probe orbit dimensions using homodyne/heterodyne measurements on pure states, or photon counters on two copies of general states. We also relate orbit dimensions to the number of directions accessible to bosonic variational circuits. Next, we study links between orbit dimensions and the resource theory of non-Gaussianity (resp. $P$-nonclassicality), and prove that free states coincide with a unique minimal-dimension orbit in the pure multimode case, under Gaussian (resp. displaced-linear) unitaries. We then extend this result to a mixed-state setting, provided that an alternative convex-roof-based definition of orbit dimensions is taken; however, we show that those fail to be fixed-mode monotones under the respective free operations. Our entire 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. Overall, this work offers a new perspective on the structure of reachable quantum states of light, which can help practitioners understand limitations and sources of expressivity and non-Gaussianity (or $P$-nonclassicality) in bosonic quantum information protocols such as quantum machine learning.