Many quantitative approaches to the dynamical scrambling of information in quantum systems involve the study of out-of-time-ordered correlators (OTOCs). In this paper we introduce an algebraic OTOC $\left(\mathcal{A}\text{−}\mathrm{OTOC}\right)$ that allows us to study information scrambling of generalized quantum subsystems under quantum channels. For closed quantum systems, this algebraic framework was recently employed to unify quantum information-theoretic notions of operator entanglement, coherence-generating power, and Loschmidt echo. The main focus of this work is to provide a natural generalization of these techniques to open quantum systems. We first show that, for unitary dynamics, the $\mathcal{A}\text{−}\mathrm{OTOC}$ quantifies a generalized notion of information scrambling, namely, between a subalgebra of observables and its commutant. For open quantum systems, on the other hand, we find a competition between the global environmental decoherence and the local scrambling of information. We illustrate this interplay by analytically studying various examples of algebras and quantum channels. To complement our analytical results, we perform numerical simulations of two paradigmatic systems: the $PXP$ model and the Heisenberg $XXX$ model, under dephasing. Our numerical results reveal connections with many-body scars and the stability of decoherence-free subspaces.

• Received 19 November 2022
• Accepted 28 March 2023

DOI:https://doi.org/10.1103/PhysRevA.107.042217