Quantum Teleportation in the Commuting Operator Framework

Abstract

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 is 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 (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 \(N\subseteq M\).