Rise and Fall, and Slow Rise Again, of Operator Entanglement under Dephasing

The operator space entanglement entropy, or simply operator entanglement (OE), is an indicator of the complexity of quantum operators and of their approximability by matrix product operators (MPOs). We study the OE of the density matrix of 1D many-body models undergoing dissipative evolution. It is expected that, after an initial linear growth reminiscent of unitary quench dynamics, the OE should be suppressed by dissipative processes as the system evolves to a simple stationary state. Surprisingly, we find that this scenario breaks down for one of the most fundamental dissipative mechanisms: dephasing. Under dephasing, after the initial rise and fall, the OE can rise again, increasing logarithmically at long times. Using a combination of MPO simulations for chains of infinite length and analytical arguments valid for strong dephasing, we demonstrate that this growth is inherent to a U(1) conservation law. We argue that in an XXZ spin model and a Bose-Hubbard model the OE grows universally as 14log2t at long times and as 2 log2 t for a Fermi-Hubbard model. We trace this behavior back to anomalous classical diffusion processes.

Automated summary

The study investigates the operator entanglement of the density matrix of 1D many-body models undergoing dissipative evolution, and finds that under the dissipative mechanism of dephasing, the operator entanglement can increase logarithmically at long times.