Manuscript Em05-B@1 · Em05-B-v1.tex · proposition · 6c050f5b

Proposition: Recovery and marginal stability

Referenced certification interfaces › Model error, decoder error, and diagnostic stability · lines 1902–1964 · proof included or linked

Em05-B@1#prop:stability, Em05-B@1#eq:purityStability, Em05-B@1#eq:occupationStability, Em05-B@1#eq:coherenceStability

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Suppose

d_{⋄} (N, N_{η}) \leq ξ .

For any fixed CPTP decoder $D$ ,

d_{⋄} (D N, id_{M}) - d_{⋄} (D N_{η}, id_{M}) \leq ξ .

The optimal errors satisfy

∣ ε_{rec} (N) - ε_{rec} (N_{η}) ∣ \leq ξ .

For any common referenced input, the output trace distance is at most $ξ$ . In particular,

∣ P_{R} - P_{R} ∣ ∣ ⟨ N_{R} ⟩ - ⟨ N_{R} ⟩ ∣ ∣ (ρ_{R} - ρ_{R})_{00, 11} ∣ \leq 2 ξ, \leq 2 ξ, \leq ξ .

For a fixed decoder, Bell fidelities differ by at most $ξ$ .

Proof. Postprocessing contraction and the reverse triangle inequality give the fixed-decoder result. Taking infima in both directions gives optimal-error stability. The referenced output statement is the channel metric definition.

Let $X = ρ_{R} - ρ_{R}$ . Since $(ρ_{R} + ρ_{R}) /2$ is an effect,

∣ Tr (ρ_{R}^{2} - ρ_{R}^{2}) ∣ = 2 Tr (\frac{ρ _{R} + ρ _{R}}{2} X) \leq 2 ξ .

The trace-zero identity $Tr X = 0$ gives

∣ Tr (N_{R} X) ∣ = ∣ Tr ((N_{R} - I_{R}) X) ∣ \leq ∥ N_{R} - I_{R} ∥_{\infty} ∥ X ∥_{1} \leq 2 ξ .

For distinct basis vectors $u, v$ , a phase-aligned Hermitian off-diagonal test operator of norm one gives

2∣ ⟨ u ∣ X ∣ v ⟩ ∣ \leq ∥ X ∥_{1} .

This proves the coherence bound. The Bell projector and postprocessing contraction give the final assertion.

Displayed formulas in this passage

Em05-B@1#eq:purityStability · line 1921
$∣ P_{R} - P_{R} ∣ ∣ ⟨ N_{R} ⟩ - ⟨ N_{R} ⟩ ∣ ∣ (ρ_{R} - ρ_{R})_{00, 11} ∣ \leq 2 ξ, \leq 2 ξ, \leq ξ .$
Em05-B@1#eq:occupationStability · line 1921
$∣ P_{R} - P_{R} ∣ ∣ ⟨ N_{R} ⟩ - ⟨ N_{R} ⟩ ∣ ∣ (ρ_{R} - ρ_{R})_{00, 11} ∣ \leq 2 ξ, \leq 2 ξ, \leq ξ .$
Em05-B@1#eq:coherenceStability · line 1921
$∣ P_{R} - P_{R} ∣ ∣ ⟨ N_{R} ⟩ - ⟨ N_{R} ⟩ ∣ ∣ (ρ_{R} - ρ_{R})_{00, 11} ∣ \leq 2 ξ, \leq 2 ξ, \leq ξ .$
unlabeled · line 1905
$d_{⋄} (N, N_{η}) \leq ξ .$
unlabeled · line 1909
$d_{⋄} (D N, id_{M}) - d_{⋄} (D N_{η}, id_{M}) \leq ξ .$
unlabeled · line 1916
$∣ ε_{rec} (N) - ε_{rec} (N_{η}) ∣ \leq ξ .$
unlabeled · line 1941
$∣ Tr (ρ_{R}^{2} - ρ_{R}^{2}) ∣ = 2 Tr (\frac{ρ _{R} + ρ _{R}}{2} X) \leq 2 ξ .$
unlabeled · line 1950
$∣ Tr (N_{R} X) ∣ = ∣ Tr ((N_{R} - I_{R}) X) ∣ \leq ∥ N_{R} - I_{R} ∥_{\infty} ∥ X ∥_{1} \leq 2 ξ .$
unlabeled · line 1959
$2∣ ⟨ u ∣ X ∣ v ⟩ ∣ \leq ∥ X ∥_{1} .$

Proposition: Recovery and marginal stability

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