  We consider quantum information tasks in an operator algebraic setting, where
we consider normal states on von Neumann algebras. In particular, we consider
subfactors $\mathfrak{N} \subset \mathfrak{M}$, that is, unital inclusions of
von Neumann algebras with trivial center. One can ask the following question:
given a normal state $\omega$ on $\mathfrak{M}$, how much can one learn by only
doing measurements from $\mathfrak{N}$? We argue how the Jones index
$[\mathfrak{M}:\mathfrak{N}]$ can be used to give a quantitative answer to
this, showing how the rich theory of subfactors can be used in a quantum
information context. As an example we discuss how the Jones index can be used
in the context of wiretap channels.
  Subfactors also occur naturally in physics. Here we discuss two examples:
rational conformal field theories and Kitaev's toric code on the plane, a
prototypical example of a topologically ordered model. There we can directly
relate aspects of the general setting to physical properties such as the
quantum dimension of the excitations. In the example of the toric code we also
show how we can calculate the index via an approximation with finite
dimensional systems. This explicit construction sheds more light on the
connection between topological order and the Jones index.
