DOI:https://doi.org/10.1103/PhysRevLett.127.229601

DOI:https://doi.org/10.1103/PhysRevLett.127.229602

We introduce deterministic state-transformation protocols between many-body quantum states that can be implemented by low-depth quantum circuits followed by local operations and classical communication. We show that this gives rise to a classification of phases in which topologically ordered states or other paradigmatic entangled states become trivial. We also investigate how the set of unitary operations

It is shown that the infinite tower of tree-level plus-helicity soft graviton symmetries in asymptotically flat 4D quantum gravity can be organized into a single chiral 2D Kac-Moody symmetry based on the wedge algebra of w1+∞, which naturally acts on the celestial sphere at null infinity. The infinite towers of soft photon or gluon symmetries also transform irreducibly under w1+∞.