What is the 'state space' of a quantum field theory called?

This is just a terminological question, not a question about reality or mathematics.

I often want to talk about state spaces in quantum field theory. For example the space of [all possible vector states in] a free scalar quantum field.

I have been told in a comment on my other question this this object is not called a "quantum field", because a "quantum field" is an operator field (or a space of operator fields). I know an operator is a kind of mapping, and takes an input. The entity I want to be able to talk about is not a mapping, it is like a vector (or it is a vector), it just exists and does not act on something else. What is the standard name for it?

Edits:

I hope this is quite a clear example: I may want to talk about the 'state of photons in the universe'. I have been told this cannot be called a quantum field, because the quantum field is an operator not a state. So I presume this cannot be called the photon field or similar? Obviously it is not a quantum field theory either because it is not a theory, it is physical. So I don't know what to call it. I have never seen a phrase like "state of photons" or "space of photon states" in use.

I think it is fair to say what I am looking for is a term that means "Hilbert space equipped with a quantum field theory interpretation" (or physical entity represented by it) based on the helpful comments and answers.