Rise and fall, and slow rise again, of operator entanglement under dephasing

TL;DR

This paper investigates the behavior of operator entanglement in 1D many-body quantum systems under dephasing, revealing a universal long-time logarithmic growth linked to conservation laws and anomalous diffusion.

Contribution

It uncovers the unexpected long-time logarithmic growth of operator entanglement under dephasing, driven by conservation laws, supported by MPO simulations and analytical arguments.

Findings

Operator entanglement initially rises and falls before growing again.

Long-time growth of OE is logarithmic, proportional to conservation laws.

Growth is linked to anomalous classical diffusion processes.

Abstract

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 (MPO). 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 $\frac{1}{4} \log_2 t$ at long times, and as $\frac{1}{2} \log_2 t$ for a Fermi-Hubbard model. We trace this behavior back to anomalous classical diffusion processes.