Manuscript En02-D@4 · En02-D-v4.tex · proposition · e19ec871

Proposition: Propagation on the full loss-closed domain

Repeated emission and the loss-closed interface · lines 757–778 · proof included or linked

En02-D@4#prop:scatter

For rail-independent coefficients,

∣ S, a ⟩ \mapsto A \subseteq S \sum r \in A \prod γ_{j r} r \in S ∖ A \prod e^{i ϕ_{j r}} 1 - γ_{j r} ∣ A, a_{A} ⟩_{D} ∣ S ∖ A, a_{S ∖ A} ⟩_{F}

is an isometry on all of $H_{loss}$ and preserves energy.

Proof. Different assignments are orthogonal and their squared amplitudes sum to one. Different inputs are distinguished by the union of occupied output modes and rail bits. Both subsets remain in $S_{K, c}$ , and their energies add to the input energy.

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unlabeled · line 760
$∣ S, a ⟩ \mapsto A \subseteq S \sum r \in A \prod γ_{j r} r \in S ∖ A \prod e^{i ϕ_{j r}} 1 - γ_{j r} ∣ A, a_{A} ⟩_{D} ∣ S ∖ A, a_{S ∖ A} ⟩_{F}$

Proposition: Propagation on the full loss-closed domain

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