# Multimode unitary channel in terms of action on characteristic function

Consider a set of $$M$$ signal modes described by the creation operators $$\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$$, and let $$\Phi_U$$ be the channel defined by the conjugation $$\Phi_U(\cdot)=U(\cdot) U^\dagger$$ by a unitary $$U$$ operator acting on the mode vector as $$U \mathbf a^\dagger U^\dagger = G \mathbf a^\dagger$$ for a symmetric, unitary matrix $$G$$ ($$U\mathbf a^\dagger U^\dagger$$ should be read as the component-wise application of the unitary to each mode).

Let $$\rho$$ be any pure multi-mode state in the corresponding Hilbert space. My question is: what is the correct way to describe the action of $$\Phi$$ on $$\rho$$ in terms of the characteristic function $$\chi_\rho(\boldsymbol\xi)=\text{Tr }\rho D_{\boldsymbol\xi}$$, where $$\boldsymbol\xi = (\xi_1,\dots,\xi_M)$$ is the displacement vector, and $$D_{\boldsymbol\xi}=\exp(\boldsymbol\xi\cdot \mathbf a^\dagger-\boldsymbol\xi^*\cdot \mathbf a)$$ is the displacement operator?

Here's a naive attempt: \begin{align} \rho' :=\Phi_U(\rho)&=\frac{1}{(2\pi)^M}\int_{\mathbb C^M} d\boldsymbol\xi \ \chi_{\rho}(\boldsymbol\xi)UD_{-\boldsymbol\xi}U^\dagger \\ &=\frac{1}{(2\pi)^M}\int_{\mathbb C^M} d\boldsymbol\xi \ \chi_{\rho}(\boldsymbol\xi)D_{-G\boldsymbol \xi} \\ &=\frac{1}{(2\pi)^M}\int_{\mathbb C^M} d\boldsymbol\xi'|\det (*)| \ \chi_{\rho}(G^{-1}\boldsymbol \xi)U_{-\boldsymbol\xi'} \end{align} which gives the new characteristic function $$\chi_{\Phi(\rho)}(\boldsymbol\xi)=|\det (*)|\chi_\rho(G^{-1}\boldsymbol\xi)$$. I have several doubts about this computation: first of all, I could only find the multi-mode generalization of the phase-space representation of $$\rho$$ in terms of the position vector $$\mathbf r$$, and secondly I'm not completely sure how to compute the change of variable factor in the complex multidimensional case (which is why I put an asterisk in the formula).

Crossposted from Physics.SE