Manuscript En07-C@1 · En07-C.tex · proposition · 3f7e9ebb

Proposition: Selected entropy and spectra do not determine recovery

Entropy, energy, erasure cost, and leakage › Selected entropy and spectral data · lines 1256–1306 · proof included or linked

En07-C@1#prop:entropy, En07-C@1#eq:entropy-good, En07-C@1#eq:entropy-bad

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Choose any finite state $σ_{K}$ , a purification $∣ χ ⟩_{K P}$ , and a logical register $L ≃ C$ . Let

R = C_{R} \otimes K, B = P \otimes L .

Define two complete isometries, with tensor factors ordered as $R$ and $B$ , by

V_{return} ∣ ψ ⟩ V_{retain} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, = ∣ 0 ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L} .

Their accessible channels are

N_{return} (X) = X \otimes σ_{K}, N_{retain} (X) = Tr (X) ∣ 0 ⟩ ⟨ 0 ∣_{C_{R}} \otimes σ_{K} .

For every pure logical input, the two accessible outputs have the same eigenvalues and the same entropy $S (σ_{K})$ . For the input $∣ 0 ⟩ ⟨ 0 ∣$ , the entire accessible density operators are equal. Nevertheless

ε_{rec} (N_{return}) = 0, ε_{rec} (N_{retain}) = q_{d} .

If $H_{R} = I_{C_{R}} \otimes H_{K}$ , every energy-measurement distribution is also equal in the two models for all logical inputs.

Proof. Both maps preserve logical inner products. The partial traces give the displayed channels. For a pure $∣ ψ ⟩$ , both accessible states are a rank-one logical state tensored with $σ_{K}$ ; therefore their nonzero spectra and entropies agree. The chosen input $∣ 0 ⟩$ makes the density operators identical. Tracing $K$ exactly decodes the return model. The retain model is a replacement channel, so Lemma lem:constant-error gives its error. A spectral effect of $H_{R} = I \otimes H_{K}$ acts as the identity on the logical factor and has the same expectation in both outputs.

Displayed formulas in this passage

En07-C@1#eq:entropy-good · line 1266
$V_{return} ∣ ψ ⟩ V_{retain} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, = ∣ 0 ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L} .$
En07-C@1#eq:entropy-bad · line 1266
$V_{return} ∣ ψ ⟩ V_{retain} ∣ ψ ⟩ = ∣ ψ ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ 0 ⟩_{L}, = ∣ 0 ⟩_{C_{R}} ∣ χ ⟩_{K P} ∣ ψ ⟩_{L} .$
unlabeled · line 1261
$R = C_{R} \otimes K, B = P \otimes L .$
unlabeled · line 1275
$N_{return} (X) = X \otimes σ_{K}, N_{retain} (X) = Tr (X) ∣ 0 ⟩ ⟨ 0 ∣_{C_{R}} \otimes σ_{K} .$
unlabeled · line 1285
$ε_{rec} (N_{return}) = 0, ε_{rec} (N_{retain}) = q_{d} .$

Proposition: Selected entropy and spectra do not determine recovery

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