



02008nas a2200181   4500008004100000245005800041210005800099260001400157300000900171490000600180520148900186100002201675700002001697700002401717700002301741700002501764856003701789       2023                        eng d00aPage curves and typical entanglement in linear optics0 aPage curves and typical entanglement in linear optics  c5/18/2023  a10170 v73 a<p>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.</p>1 aIosue, Joseph, T.1 aEhrenberg, Adam1 aHangleiter, Dominik1 aDeshpande, Abhinav1 aGorshkov, Alexey, V.  uhttps://arxiv.org/abs/2209.0683801855nas a2200145   4500008004100000245007300041210006900114260001500183520138800198100002201586700001901608700002001627700002501647856003701672       2023                        eng d00aProjective toric designs, difference sets, and quantum state designs0 aProjective toric designs difference sets and quantum state desig  c11/22/20233 a<p>Trigonometric cubature rules of degree t are sets of points on the torus over which sums reproduce integrals of degree t monomials over the full torus. They can be thought of as t-designs on the torus. Motivated by the projective structure of quantum mechanics, we develop the notion of t-designs on the projective torus, which, surprisingly, have a much more restricted structure than their counterparts on full tori. We provide various constructions of these projective toric designs and prove some bounds on their size and characterizations of their structure. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures (SIC-POVMs) and complete sets of mutually unbiased bases (MUBs) (which are conjectured to relate to finite projective geometry) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense Btmodm sets. We also use projective toric designs to construct families of quantum state designs. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information.</p>1 aIosue, Joseph, T.1 aMooney, T., C.1 aEhrenberg, Adam1 aGorshkov, Alexey, V.  uhttps://arxiv.org/abs/2311.13479