Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuousvariable quantum computation as well as to nearterm 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 Rényi2 Page curve (the average Rényi2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi2 entropy by studying its variance. Using the aforementioned results for the Rényi2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.
more » « less Award ID(s):
 2120757
 NSFPAR ID:
 10505877
 Publisher / Repository:
 Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften
 Date Published:
 Journal Name:
 Quantum
 Volume:
 7
 ISSN:
 2521327X
 Page Range / eLocation ID:
 1017
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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