Manuscript Psy02-A@1 · Psy02-A.tex · proposition · d92ce402

Proposition: Attenuation moment and displacement identities

Environmental dynamics and the two-budget principle › An explicit attenuator and its separate budgets · lines 1121–1147 · proof included or linked

Psy02-A@1#prop:attenuation

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For every finite-number input $ρ$ ,

n (A_{η} (ρ)) = η n (ρ) + (1 - η) n_{B} .

Furthermore,

A_{η} (U_{ϵ} ρ U_{ϵ}^{†}) = W (η d / ϵ) A_{η} (ρ) W (η d / ϵ)^{†} .

Proof. Expand $a_{j, out}^{†} a_{j, out}$ . The cross terms vanish because the bath is independent and centered. Sum over $j$ . A displacement of the system transforms into a system displacement of amplitude $η$ and a bath displacement. The latter disappears under partial trace.

Displayed formulas in this passage

unlabeled · line 1124
$n (A_{η} (ρ)) = η n (ρ) + (1 - η) n_{B} .$
unlabeled · line 1130
$A_{η} (U_{ϵ} ρ U_{ϵ}^{†}) = W (η d / ϵ) A_{η} (ρ) W (η d / ϵ)^{†} .$

Proposition: Attenuation moment and displacement identities

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