Quantum teleportation in the commuting operator framework

We introduce a notion of teleportation scheme between subalgebras of semi-finite von Neumann algebras in the commuting operator model of locality. Using techniques from subfactor theory, we present unbiased teleportation schemes for relative commutants $N'\cap M$ of a large class of finite-index inclusions $N\subseteq M$ of tracial von Neumann algebras, where the unbiased condition means that no information about the teleported observables are contained in the classical communication sent between the parties. For a large class of subalgebras $N$ of matrix algebras $M_n(\mathbb{C})$, including those relevant to hybrid classical/quantum codes, we show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(\mathbb{C})$ over $N'$, generalising work of Werner. Combining our techniques with those of Brannan-Ganesan-Harris, we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions $N\subseteq M$.