Projector matrix product operators, anyons and higher relative commutants of subfactors

A bi-unitary connection in subfactor theory of Jones producing a finite depth gives 4-tensor appearing recent work Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete on 2-dimensional topological order and anyons. In their work, they have special projection called projector matrix product operator. We prove that the range this length k is naturally identified with k-th higher relative commutant arising from connection. This further between theory.