Manuscript Hawk04-C@1 · Hawk04-C-v1.tex · proposition · 94d46352

Proposition: Identical thermal observations, different recovery

Row 6: thermality, classical records, and information loss › Selected thermal data do not determine recovery · lines 1405–1448 · proof included or linked

Hawk04-C@1#prop:thermal

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Fix a finite Gibbs state

τ_{T} = \frac{e ^{- β H_{T}}}{Tr ( e ^{- β H_{T}} )}

and a purification $∣ Θ ⟩_{T T^{'}}$ . Let $A ≃ B_{1} ≃ C$ , and choose

R = T \otimes A, B = T^{'} \otimes B_{1} .

After fixed factor reordering, define isometries by

V_{rec} ∣ ψ ⟩ V_{hide} ∣ ψ ⟩ = ∣ Θ ⟩_{T T^{'}} ∣ ψ ⟩_{A} ∣ 0 ⟩_{B_{1}}, = ∣ Θ ⟩_{T T^{'}} ∣ 0 ⟩_{A} ∣ ψ ⟩_{B_{1}} .

Every observable on $T$ has the same expectation in the two models, for every logical input. Nevertheless,

ε_{rec} (N_{rec}) = 0, ε_{rec} (N_{hide}) = 1 - d^{- 2} .

Proof. The factor $Θ$ and the blank vector are fixed and normalized, so both maps are isometries. Their accessible channels are

N_{rec} (X) = τ_{T} \otimes X_{A}, N_{hide} (X) = Tr (X) τ_{T} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{A} .

The $T$ -reductions coincide. Tracing $T$ recovers the first input exactly. The second channel is a replacement, so Lemma lem:constant applies.

Displayed formulas in this passage

unlabeled · line 1418
$V_{rec} ∣ ψ ⟩ V_{hide} ∣ ψ ⟩ = ∣ Θ ⟩_{T T^{'}} ∣ ψ ⟩_{A} ∣ 0 ⟩_{B_{1}}, = ∣ Θ ⟩_{T T^{'}} ∣ 0 ⟩_{A} ∣ ψ ⟩_{B_{1}} .$
unlabeled · line 1408
$τ_{T} = \frac{e ^{- β H_{T}}}{Tr ( e ^{- β H_{T}} )}$
unlabeled · line 1414
$R = T \otimes A, B = T^{'} \otimes B_{1} .$
unlabeled · line 1429
$ε_{rec} (N_{rec}) = 0, ε_{rec} (N_{hide}) = 1 - d^{- 2} .$
unlabeled · line 1439
$N_{rec} (X) = τ_{T} \otimes X_{A}, N_{hide} (X) = Tr (X) τ_{T} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{A} .$

Proposition: Identical thermal observations, different recovery

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