Operational Quantum Mereology and Minimal Scrambling

In this paper we will attempt to answer the following question: what are the natural quantum subsystems which emerge out of a system's dynamical laws? To answer this question we first define generalized tensor product structures (gTPS) in terms of observables, as dual pairs of an operator subalgebra $\cal A$ and its commutant. Second, we propose an operational criterion of minimal information scrambling at short time scales to dynamically select gTPS. In this way the emergent subsystems are those which maintain the longest informational identity. This strategy is made quantitative by defining a Gaussian scrambling rate in terms of the short-time expansion of an algebraic version of the Out of Time Order Correlation (OTOC) function i.e., the $\cal A$-OTOC. The Gaussian scrambling rate is computed analytically for physically important cases of general division into subsystems, and is shown to have an intuitive and compelling physical interpretation in terms of minimizing the interaction strength between subsystems.