Approximate 2-localization of random matrices: The random matrix universe

Quantum many-body systems are endowed with a tensor product structure. This structure is essentially inherited from probability theory, where the probability of two independent events is the product of the probabilities. The tensor product structure of a Hamiltonian thus gives a natural decomposition of the system into smaller subsystems. Considering a particular Hamiltonian and a particular tensor product structure, one can ask: is there a basis in which this Hamiltonian has this desired tensor product structure? In general such an exact structure does not exist, however we will show (numerically) that large random matrices are approximately 2-local in a carefully chosen basis. These results suggest a mechanism for the emergence of locality from quantum theory itself.