Minimal Areas from entangled matrices

The geometrization of quantum information lies at the core of holography and of quantum gravity more broadly. This principle is exemplified by the Ryu-Takayanagi (RT) formula equating entanglement entropy to a minimal surface in a dual geometry. In this talk I will illustrate how the entanglement entropy of relational subsystems in theories of matrix quantum mechanics can give rise to a minimization and counting problem exhibiting many similarities to the RT formula. In particular, in states where non-commutative geometry emerges from semiclassical matrices, the subsystem determines a reduced state which is (approximately) an incoherent sum of density matrices corresponding to distinct spatial subregions, the areas of which count the dimension of maximally entangled edge modes. I will further show how this sum can be dominated by a subregion of minimal boundary area. Central to this result is a notion of coarse-graining that controls the proliferation of highly curved and disconnected non-geometric subregions in the sum.