I am reading a paper about Gaussian quantum states and the mathematical formalism used to describe them. At one point the authors say "An important example of a Gaussian state is the maximally entangled state $\Phi$. In their endnotes, they then note that
"Although this maximally entangled state does not belong to the Hilbert space, it can always be considered as a limit of a proper pure state."
Why does the maximally entangled state not belong to the Hilbert space? Is it in some sense ill-defined as a pure state?