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 subset of M of tracial von Neumann algebras, where the unbiased condition means that no information about the teleported observables is contained in the classical communication sent between the parties. For a large class of subalgebras N of matrix algebras Mn(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 Mn(C) over N', generalising work of Werner (J Phys A 34(35):7081-7094, 2001). Combining our techniques with those of Brannan-Ganesan-Harris (J Math Phys 63(11): 112204, 2022) we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions N subset of M.