Tensor Product Structure Geometry under Unitary Channels

Locality is typically defined with respect to a tensor product structure (TPS) which identifies the local subsystems of the quantum system. In this paper, we investigate a simple geometric measure of operator spreading by quantifying the distance of the space of
local operators from itself evolved under a unitary channel. We show that this TPS distance is related to the scrambling properties of
the dynamics between the local subsystems and coincides with the entangling power of the dynamics in the case of a symmetric bipartition. For Hamiltonian evolutions at short times, the characteristic timescale of the TPS distance depends on scrambling rates determined by the strength of interactions between the local subsystems. Beyond this short-time regime, the behavior of the TPS
distance is explored through numerical simulations of prototypical models exhibiting distinct ergodic properties, ranging from quantum
chaos and integrability to Hilbert space fragmentation and localization.