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Here, linearity refers to the Hamiltonian, which is linear/nonlinear with respect to creation and annihilation operators and passive means photon-number conserving.
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21
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0012317010
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note
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Note that the direct sum corresponds to a split between modes rather than to a position/momentum split.
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note
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Here we have used that for any symplectic matrix det[S] = 1, and admissible covariance matrices satisfy det[Γ] ≥ 1.
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