Let $b_{1,2}$ be bosonic operators that satisfy $[b_i, b_i^\dagger]=1$. Two boson operators give the Hilbert space $\mathcal H_1 \otimes \mathcal H_2$ in the tensor product form.
Consider the Gaussian state $$|\Phi\rangle = e^{\lambda (b_1^\dagger b_2^\dagger - b_1b_2)} |0\rangle,$$ where $|0\rangle$ is the vaccum and $\lambda\in\mathbb R$. Taking partial trace to subsystem labeled by 2, we obtain $\rho_1 = \operatorname{Tr}_2(|\Phi\rangle \langle \Phi|)$.
Question: What is the purity $\operatorname{Tr}(\rho_1^2)$?
Even though the state is Gaussian, it seems that the purity computation is not easy. I tried to expand $$|\Phi\rangle \langle \Phi| = \sum_{n,m\geq 0} \frac{(-1)^m}{n!m!} \lambda^{n+m} (b_1^\dagger b_2^\dagger -b_1b_2)^n |0\rangle \langle 0| (b_1^\dagger b_2^\dagger -b_1b_2)^m.$$ Since there are too many terms, taking partial trace seems not easy.
I am also interested in to what extent we can compute the entanglement measures explicitly, for Gaussian states.