  The operator growth hypothesis (OGH) is a technical conjecture about the
behaviour of operators -- specifically, the asymptotic growth of their Lanczos
coefficients -- under repeated action by a Liouvillian. It is expected to hold
for a sufficiently generic closed many-body system. When it holds, it yields
bounds on the high frequency behavior of local correlation functions and
measures of chaos (like OTOCs). It also gives a route to numerically estimating
response functions. Here we investigate the generalisation of OGH to open
quantum systems, where the Liouvillian is replaced by a Lindbladian. For a
quantum system with local Hermitian jump operators, we show that the OGH is
modified: we define a generalisation of the Lanczos coefficient and show that
it initially grows linearly as in the original OGH, but experiences
exponentially growing oscillations on scales determined by the dissipation
strength. We see this behavior manifested in a semi-analytically solvable model
(large-q SYK with dissipation), numerically for an ergodic spin chain, and in a
solvable toy model for operator growth in the presence of dissipation (which
resembles a non-Hermitian single-particle hopping process). Finally, we show
that the modified OGH connects to a fundamental difference between Lindblad and
closed systems: at high frequencies, the spectral functions of the former decay
algebraically, while in the latter they decay exponentially. This is an
experimentally testable statement, which also places limitations on the
applicability of Lindbladians to systems in contact with equilibrium
environments.
