



01855nas 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