Page curves and typical entanglement in linear optics

Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Renyi-2 Page curve (the average Renyi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the cor-responding Page correction (the average infor-mation of the subsystem) in certain squeez-ing regimes. We then prove various results on the typicality of entanglement as measured by the Renyi-2 entropy by studying its vari-ance. Using the aforementioned results for the Renyi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typical-ity as measured by the von Neumann entropy. Our main proofs make use of a symmetry prop-erty obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we pro-pose future research directions where this sym-metry might also be exploited. We conclude by discussing potential applications of our re-sults and their generalizations to Gaussian Bo-son Sampling and to illuminating the relation-ship between entanglement and computational complexity.