Operator Entanglement in Interacting Integrable Quantum Systems: the Case of the Rule 54 Chain

Abstract

In a many-body quantum system, local operators in Heisenberg picture $O(t) = e^{i H t} O e^{-i H t}$ spread as time increases. Recent studies have attempted to find features of that spreading which could distinguish between chaotic and integrable dynamics. The operator entanglement - the entanglement entropy in operator space - is a natural candidate to provide such a distinction. Indeed, while it is believed that the operator entanglement grows linearly with time $t$ in chaotic systems, numerics suggests that it grows only logarithmically in integrable systems. That logarithmic growth has already been established for non-interacting fermions, however progress on interacting integrable systems has proved very difficult. Here, for the first time, a logarithmic upper bound is established rigorously for all local operators in such a system: the `Rule 54' qubit chain, a model of cellular automaton introduced in the 1990s [Bobenko et al., CMP 158, 127 (1993)], recently advertised as the simplest representative of interacting integrable systems. Physically, the logarithmic bound originates from the fact that the dynamics of the models is mapped onto the one of stable quasiparticles that scatter elastically; the possibility of generalizing this scenario to other interacting integrable systems is briefly discussed.