In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems, as is the case for the vacua of quantum field theories or ground states in infinite spin chains. But are there different kinds of infinite entanglement?

I will review a series of recent works studying operational tasks that can distinguish different forms of infinite entanglement. Formulated in the framework of von Neumann algebras — the natural way of modeling (fixed sectors of) infinite systems — these works uncover a surprisingly strong connection:

The type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with operational entanglement properties. For instance, the Connes classification of type III factors corresponds to the system's capability at the operational task of "embezzling" entanglement from the system.

This promotes the type classification from mere algebraic bookkeeping to a classification of infinite quantum systems based on their entanglement properties.

Joint work with Alexander Stottmeister, Reinhard F. Werner, and Henrik Wilming.