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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

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    $\begingroup$ Please crosslink crossposts. $\endgroup$ Mar 15 at 0:49

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