Most of the recent work on quantum information theory has been carried out in the setting of finite dimensional Hilbert spaces. While this is certainly sufficient for most of the central questions in the field, there are many reasons to explore what happens if one goes beyond this framework.

(1) Real quantum systems come with a natural description in infinite dimensional Hilbert spaces. If one wants to analyze how the 'good' qubit variables are embedded in the system, and what interplay with with the rest of the system might lead to decoherence, the basic framework has to allow for infinite dimensional systems. Moreover, if the 'rest of the system' comprises the radiation field, as in decoherence due to spontaneous emission, the whole system is not just one in an infinite dimensional Hilbert space, but has infinitely many degrees of freedom.

(2) One might hope that the new understanding gained from quantum information theory might be helpful for analyzing some traditional problems involving infinitely many degrees of freedom, notably in statistical mechanics and quantum field theory. These disciplines have also developed the mathematical tools for dealing with infinitely many degrees of freedom, however, without including entanglement so far.

(3) Threshold behaviour of distillation processes may be more than superficially related to phase transitions, which become mathematically sharp only in the thermodynamic limit.

(4) Systems with actually infinite entanglement describe in a sharp way an asymptotic direction in the study of entangled finite systems. In some respects the limit may be easier to analyze than large finite systems. In particular, the entangled state appearing in the 1935 paper of Einstein, Rosen and Podolsky can be analyzed rigorously as such an asymptotic object.

This talk begins with a brief look at states for which all the usual entanglement measures diverge. In particular, if the distillible entanglement is infinite, we can extract singlets from multiple copies of the state, at a rate which goes to infinity as we take more and more copies. In some sense each system thus contains infinite entanglement. But then one might try to do the same thing with just one copy of the system. Our first main result is that this is impossible in the standard framework of quantum mechanics:

Theorem: There is no density operator on a tensor product of (infinite dimensional) Hilbert spaces such that we can extract from it arbitrarily highly entangled (finite dimensional) states by local quantum operations and classical communication.

In the sequel we show how states with 'infinite one-shot distillible entanglement' can nevertheless by constructed. There are two ways for doing this, because there are two assumptions in the No-Go Theorem, which can be relaxed without endangering the physical interpretation of the quantum formalism.

On the one hand we can relax the requirement that states are given by density operators, and allow instead more general expectation value functionals, assigning to every positive operator a positive expectation. Of course, in finite dimension it is a theorem, that all such functionals can be represented as the trace with a density operator. In infinite dimension, however, this representation theorem requires a continuity property, which can be relaxed. This amounts to using so-called singular states. Since the space of singular states is compact, limiting situations of all sorts can be accomodated. In particular, we show what can be said about the original EPR state, in which the canonical variables $Q_1-Q_2$ and $P_1+P_2$ have sharp values, and which can be seen as the limit of infinitely squeezed two-mode Gaussians.

On the other hand, the complete specification of a singular state is impossible, since already the proof that such states exist requires the Axiom of Choice. Anyhow, what one is interested in are usually not all observables, but only a relatively small subset. We follow quantum field theory in taking the sets of observables defining each subsystem as von Neumann algebras. In this language we can still say what it means that arbitrary measurements of Alice and Bob can be combined: their respective observable algebras have to commute. Again, in the finite dimensional context we can construct from this statement a tensor product decomposition of the underlying Hilbert space. This fails in the infinite dimensional case, and we can readily construct bipartite systems corresponding to infinitely many singlets. It turns out that there is an essentially unique pair of observables with a state, which is the analog of maximally entangled states of finite systems. One of the hallmarks of this structure is that all normal states violate the Bell-CHSH inequalities maximally. An example is the vacuum of quantum field theory. Vacuum fluctuations are thus infinitely non-classical, although this is strictly speaking an ultraviolet effect appearing only when Alice and Bob are spacelike separated, but with zero distance.