I think that we can write a state in Quantum Mechanics like: $$ \int dx\lambda _x \vert x\rangle\langle x\vert $$ and when we talk about QFT, we would replace integral with functional integrals: $$ \int D\psi \lambda _\psi \vert\psi\rangle\langle\psi\vert $$
When we look at a mixed state in QM in following form: $$\omega = \int d\mu (\gamma )\lambda _\gamma \mathop{\omega _\gamma}$$ where, $d\mu(\gamma )$ is Haar $\operatorname{U}(1)$ invariant measure, how should we go from QM to QFT. I think that one difference should also lie in the point that we do not have fields any longer but operators. How would we represent it in QFT? Where exactly will the difference lie?