Quantum teleportation in the commuting operator framework

TL;DR

This paper develops a framework for quantum teleportation within the commuting operator model of locality, utilizing subfactor theory to construct unbiased schemes for von Neumann algebra subalgebras, and applies these to quantum graph coloring problems.

Contribution

It introduces a new notion of teleportation schemes in the commuting operator framework and characterizes their structure using subfactor theory, extending previous work on quantum teleportation.

Findings

Unbiased teleportation schemes are constructed for relative commutants in finite-index von Neumann algebra inclusions.

Any tight teleportation scheme for certain subalgebras arises from an orthonormal unitary Pimsner-Popa basis.

Quantum chromatic numbers are computed for quantum graphs from finite-dimensional algebra inclusions.

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 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 $N\subseteq M$.