The overlap of two Gaussian states According to e.g. Serafini (Quantum Continuous Variables), the Hilbert-Schmidt product ('overlap') of two multimode Gaussian states $\rho_1,\rho_2$ is
$$\text{Tr}[\rho_1\rho_2]=|\langle\psi_1|\psi_2\rangle|^2=\frac{2^n}{\sqrt{\text{Det}(\sigma_1+\sigma_2)}}e^{(r_1-r_2)^T(\sigma_1+\sigma_2)^{-1}(r_1-r_2)},$$
where $r_{1,2}$ are the displacements in phase space and $\sigma_1,\sigma_2$ the covariance matrices.
My question: is there a similar formula for $\langle\psi_1|\psi_2\rangle$ itself, retaining information on their relative phase?
 A: There cannot be such an expression because the covariance matrix and displacement don't contain the relative phase information. This is easy to see, since they are computed from the reduced density matrix (which does not depend on the phase of the state).
A way around this can be to include a third reference state and $\vert\chi\rangle$ and consider the overlap
$$
\langle \psi_1|\psi_2\rangle\langle\psi_2\vert\chi\rangle\langle\chi\vert\psi_1\rangle
$$
which only depends on the density matrices, and is thus a function of the covariance matrices & displacement.
Note, however, that also this quantity is (necessarily!) invariant under changing phases, so whether it gives you the desired information depends on the scenario you consider.
A: If the Gaussians are not too far from coherent states, a useful approximation is obtained by calculating the phasefactors of a coherent state, for which analytical formulas do exist (this comes down to taking into accound the displacement but not the squeezing/covariance matrix). According to e.g. the book Quantum Noise (Gardiner-Zoller), the scalar product of two single-mode coherent states is
$$\langle\alpha|\beta\rangle=\exp(\alpha^*\beta-\frac{1}{2}|\alpha|^2-\frac{1}{2}|\beta|^2),$$
from which one immediately gets
$$\frac{\langle\alpha|\beta\rangle}{|\langle\alpha|\beta\rangle|}=\exp(\frac{\alpha^*\beta-\beta^*\alpha}{2})$$ for the phasefactor.
As a coherent state factorizes over all modes, the overall phasefactor under this approximation is the product of the phasefactors over these individual modes.
