Manuscript Hawk05-A@1 · Hawk05-A.tex · proposition · 449f85bb

Proposition: Thermal erasure requires additional assumptions

Finite recurrence and the three ledgers › Three different kinds of data · lines 1679–1720 · proof included or linked

Hawk05-A@1#prop:landauer

Let $M$ initially be uncorrelated with a reservoir $L$ in

γ_{L} = e^{- β H_{L}} / Z, β = (k_{B} T)^{- 1} .

Suppose the complete $M L$ process is unitary, and define reservoir heat $Q_{L} = Tr H_{L} (ρ_{L}^{'} - γ_{L})$ . Then, with entropies and relative entropy $D_{2}$ in bits,

β Q_{L} = ln 2 [S (ρ_{M}) - S (ρ_{M}^{'}) + I (M : L)_{ρ^{'}} + D_{2} (ρ_{L}^{'} ∥ γ_{L})] .

In particular,

Q_{L} \geq k_{B} T ln 2 [S (ρ_{M}) - S (ρ_{M}^{'})] .

Proof. Unitary preservation of the initial product entropy gives

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

The Gibbs expression gives

ln 2 D_{2} (ρ_{L}^{'} ∥ γ_{L}) = β Q_{L} - ln 2 [S (ρ_{L}^{'}) - S (γ_{L})] .

Combine the equations. Mutual information and relative entropy are nonnegative.

Displayed formulas in this passage

unlabeled · line 1682
$γ_{L} = e^{- β H_{L}} / Z, β = (k_{B} T)^{- 1} .$
unlabeled · line 1689
$β Q_{L} = ln 2 [S (ρ_{M}) - S (ρ_{M}^{'}) + I (M : L)_{ρ^{'}} + D_{2} (ρ_{L}^{'} ∥ γ_{L})] .$
unlabeled · line 1699
$Q_{L} \geq k_{B} T ln 2 [S (ρ_{M}) - S (ρ_{M}^{'})] .$
unlabeled · line 1707
$S (ρ_{L}^{'}) - S (γ_{L}) = S (ρ_{M}) - S (ρ_{M}^{'}) + I (M : L)_{ρ^{'}} .$
unlabeled · line 1713
$ln 2 D_{2} (ρ_{L}^{'} ∥ γ_{L}) = β Q_{L} - ln 2 [S (ρ_{L}^{'}) - S (γ_{L})] .$

Proposition: Thermal erasure requires additional assumptions

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