A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory

Abstract Unitary dynamics with a strict causal cone (or “light cone”) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension completely classified by an index theory. Physical systems often exhibit only approximate cones; Hamiltonian evolutions on lattice satisfy Lieb–Robinson bounds rather than locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that theory is robust and extends one-dimensional ALPUs. As consequence, we achieve converse bounds: any ALPU zero can be exactly generated some time-dependent, quasi-local constant time. For special case finite chains open boundaries, unitary satisfying bound may such Hamiltonian. also discuss results stability operator algebras which independent interest.