# Operator Entanglement in Interacting Integrable Quantum Systems: the Case of the Rule 54 Chain. (arXiv:1901.04521v2 [cond-mat.stat-mech] UPDATED)

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.