Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · proposition · 39b16f67

Proposition: A specified finite thermal balance

Stage 3: energy, erasure cost, and a conditional mass law › Three independent ledgers · lines 1351–1381 · proof included or linked

Hawk04-D@1#prop:thermal-balance, Hawk04-D@1#eq:Landauer

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Let a memory and finite reservoir start in $ρ_{M} \otimes γ_{L}$ , with $γ_{L} = e^{- β H_{L}} / Tr e^{- β H_{L}}$ , $β > 0$ . After a specified unitary, define

Q_{L} = Tr (H_{L} (ρ_{L}^{'} - γ_{L})), Δ S_{M} = S (ρ_{M}) - S (ρ_{M}^{'}) .

Using natural logarithms,

β Q_{L} = Δ S_{M} + I (M : L)_{ρ^{'}} + D (ρ_{L}^{'} ∥ γ_{L}) \geq Δ S_{M} .

Proof. Unitary entropy preservation and the initial product state give

S (ρ_{L}^{'}) - S (γ_{L}) = Δ S_{M} + I (M : L)_{ρ^{'}} .

Using $lo g γ_{L} = - β H_{L} - lo g Z$ gives

D (ρ_{L}^{'} ∥ γ_{L}) = β Q_{L} - [S (ρ_{L}^{'}) - S (γ_{L})] .

Combine the identities and use nonnegativity.

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Hawk04-D@1#eq:Landauer · line 1362
$β Q_{L} = Δ S_{M} + I (M : L)_{ρ^{'}} + D (ρ_{L}^{'} ∥ γ_{L}) \geq Δ S_{M} .$
unlabeled · line 1357
$Q_{L} = Tr (H_{L} (ρ_{L}^{'} - γ_{L})), Δ S_{M} = S (ρ_{M}) - S (ρ_{M}^{'}) .$
unlabeled · line 1372
$S (ρ_{L}^{'}) - S (γ_{L}) = Δ S_{M} + I (M : L)_{ρ^{'}} .$
unlabeled · line 1376
$D (ρ_{L}^{'} ∥ γ_{L}) = β Q_{L} - [S (ρ_{L}^{'}) - S (γ_{L})] .$

Proposition: A specified finite thermal balance

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