The optimal unitary dilation for bosonic Gaussian channels

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Namely, if ρ ^ E = | ψ E ⟩ ​ ⟨ ψ E | {{\hat{\rho}}}_{E}=|\psi_{E}\rangle\langle\psi_{E}| , then Eq. ( 1

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Note that for a positive semi-definite matrix M M we have ker ​ [ M ] = { w ∈ ℂ d : w † ​ M ​ w = 0 } \mbox{ker}[M]=\left\{w\in\mathbb{C}^{d}:w^{{\dagger}}Mw=0\right\}

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Given Π \Pi the projector on the support of Y Y and defined the invertible matrix Y ¯ := Y + ( 𝟙 − Π ) \bar{Y}:=Y+(\openone-\Pi) , the MP-inverse of Y Y is defined as Y ⊖ 1 := Π ​ Y ¯ − 1 ​ Π Y^{\ominus 1}:=\Pi\bar{Y}^{-1}\Pi (see also Ref. [ 22 ] )

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