A Frobenius-Descent Formalism for the Motivic Filtration on K-Theory of Analytic Adic Stacks

The quantification of entanglement in quantum many-body systems remains a central challenge in condensed matter physics and quantum information theory. While typical pure states in large Hilbert spaces are known to be nearly maximally entangled, states relevant to physical systems, such as ground states of local Hamiltonians, often exhibit more structured entanglement properties. This paper investigates the entanglement entropy of bipartite quantum systems described by structured random matrix ensembles. We introduce a novel framework that rigorously connects the asymptotic expected von Neumann entropy of a subsystem to the operator-valued free entropy, a sophisticated concept from Voiculescu's free probability theory. We consider a model where a quantum system is described by the ground state of a Hamiltonian composed of interacting random matrices acting on distinct subsystems. By constructing an appropriate operator-valued non-commutative probability space, we demonstrate that the families of random matrices become asymptotically free with amalgamation over a matrix algebra representing the subsystem structure. Our main result provides a closed-form expression for the limiting entanglement entropy in terms of the operator-valued free entropy of the corresponding semicircular elements. This result establishes a direct link between the algebraic structure of non-commutative random variables and a fundamental physical quantity. As a key consequence, we show that for certain classes of interactions, this framework naturally gives rise to volume-law entanglement, offering a new analytical perspective on the behavior of complex, chaotic quantum systems.