Scrambling of algebras in open quantum systems

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