Page curves and typical entanglement in linear optics

Joseph T. Iosue1,2, Adam Ehrenberg1,2, Dominik Hangleiter2,1, Abhinav Deshpande3, and Alexey V. Gorshkov1,2

1Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
2Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA

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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 Rényi-2 Page curve (the average Rényi-2 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 Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 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.

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What gives quantum computers an advantage over their classical counterparts? It is known that entanglement is necessary for quantum advantage, but a quantitative link between entanglement and complexity is missing. The first step toward building such a link is understanding the entanglement of quantum states that are hard to classically simulate. Such a study has not been done even for the first sampling scheme shown to have a quantum advantage, namely output states of linear optical circuits. In this work, we address this by characterizing the typical entanglement of such states.

Specifically, we study bipartite entanglement within quantum states generated by random linear optical circuits acting on specially prepared inputs. We derive an exact formula for the average entanglement and prove that, in certain regimes, the probability that a random state’s entanglement deviates from the average vanishes asymptotically in the system size. Our results are obtained through a combination of methods coming from quantum optics and quantum information as well as from a novel technique that we develop based on a powerful symmetry present in the entanglement structure. We further propose how this new technique may be useful to study bipartite entanglement in different settings.

These results provide a stepping stone to a better understanding of the typical behavior of random linear optical circuits and the certification of quantum advantage in a linear optics sampling experiment.

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