Operator Space Entangling Power of Quantum Dynamics and Local Operator Entanglement Growth in Dual-Unitary Circuits

TL;DR

This paper introduces a measure for the entangling power of quantum operators, linking it to scrambling and operator entanglement growth, with implications for understanding quantum chaos and information dynamics.

Contribution

It defines operator space entangling power, relates it to scrambling, and analyzes its growth in dual-unitary circuits and Hamiltonian dynamics.

Findings

Operator space entangling power is linked to quantum scrambling.

Short-time growth rate matches Gaussian scrambling rate.

Average local operator entanglement growth shows two regimes.

Abstract

Operator entanglement is a well-established measure of operator complexity across a system bipartition. In this work, we introduce a measure for the ability of a unitary channel to generate operator entanglement, representing an operator-level generalization of the state-space entangling power. This operator space entangling power is demonstrated to be linked to the scrambling properties of the unitary channel via the recently introduced concept of mutual averaged non-commutativity of quantum operator algebras. An upper bound for the operator space entangling power is identified, corresponding to unitary channels with scrambling properties akin to those of typical unitaries. Additionally, for Hamiltonian dynamics, we find that the short-time growth rate of the operator space entangling power matches the Gaussian scrambling rate of the bipartite out-of-time-order-correlator, establishing a direct link between information scrambling and operator entanglement generation for short time scales. Finally, we examine the average growth of local operator entanglement across a symmetric bipartition of a spin-chain. For dual-unitary circuits, a combination of analytical and numerical investigations demonstrates that the average growth of local operator entanglement exhibits two distinct regimes in relation to the operator space entangling power of the building-block gate.