In field theory, by quantizing a dirac field, we can obtain a creation operator for a single electron of definite momentum, of definite spin up or down, these respectively are: $$a^\dagger_{+}(p)|0\rangle, {a^\dagger}_{-}(p)|0\rangle$$ Where we've defined the former to create a spin +1/2 electron, the latter to create a spin -1/2 electron. By addition and repeated-application of these creation operators we can write down a state of any number of particles, each having any superposition of spins. : $$ \int dp f(p)\prod_{i=0}^n(\alpha_ia^\dagger_{i+}(p)+\beta_i{a^\dagger}_{i-}(p))|0 \rangle $$ Where of course $i$ labels the particle, and $a,b,f$ are some distributions.
Question: For a given field theory, how does one write down a creation operator for a pair of entangled particles? (say electrons in a Dirac theory of spinors)
In quantum mechanics, an entangled state is one which lives in a tensor product Hilbert space, but does not have a tensor product decomposition. Since Fock space is essentially built up with a bunch of tensor products of Hilbert spaces, it doesn't seem unreasonable to demand that it contains such entangled states. But how does one explicitly write down such a state?