  For states in infinite dimensional Hilbert spaces entanglement quantities
like the entanglement of distillation can become infinite. This leads naturally
to the question, whether one system in such an infinitely entangled state can
serve as a resource for tasks like the teleportation of arbitrarily many
qubits. We show that appropriate states cannot be obtained by density operators
in an infinite dimensional Hilbert space. However, using techniques for the
description of infinitely many degrees of freedom from field theory and
statistical mechanics, such states can nevertheless be constructed rigorously.
We explore two related possibilities, namely an extended notion of algebras of
observables, and the use of singular states on the algebra of bounded
operators. As applications we construct the essentially unique infinite
analogue of maximally entangled states, and the singular state used
heuristically in the fundamental paper of Einstein, Rosen and Podolsky.
