Manuscript Em03-H@8 · Em03-H-v8.tex · section · 91e9e3bd

Metrics and standard facts

Scope, evidence, and the version-8 revision › Metrics and standard facts · lines 307–359

The message is $M$ , reference $Q$ , boundary $B$ , and bath $R$ . Accessible bath modes form $A$ . A non-message factor $K$ may have an inaccessible purifying register $G$ .

Unsubscripted norms are operator norms. Subscripts $1$ and $F$ denote trace and Frobenius norms. Set

d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .

All channel distances retain arbitrary references. For an isometry $S$ , $Ad_{S} (X) = S X S^{†}$ , and $R_{τ} (X) = tr (X) τ$ . Matrix units are $E_{α β} = ∣ α ⟩ ⟨ β ∣$ .

For a qubit channel,

ε_{rec} (N) = D CPTP in f d_{⋄} (DN, id_{2}) .

Let $∣ Φ ⟩ = (∣ 00 ⟩ + ∣ 11 ⟩) / 2$ . Define squared target-overlap fidelities

F_{e} (Λ) = ⟨ Φ ∣ (id \otimes Λ) (∣ Φ ⟩ ⟨ Φ ∣) ∣ Φ ⟩,

F_{wc} (Λ) = ρ_{M} in f ⟨ ψ_{ρ} ∣ (id \otimes Λ) (∣ ψ_{ρ} ⟩ ⟨ ψ_{ρ} ∣) ∣ ψ_{ρ} ⟩ .

Root fidelity $f (ρ, σ) = ρ σ_{1}$ satisfies $f \geq 1 - d_{tr} (ρ, σ)$ . Its maximal purification overlap equals $f$ , with sufficiently enlarged purifying spaces. The diamond norm of a difference of finite channels can be optimized on referenced density operators.

These conventions and facts are standard Watrous,FuchsVandeGraaf. Recovery and information–disturbance frameworks are established BenyOreshkov,KSW. A channel error at most $e$ against the identity implies both target overlaps are at least $1 - e$ , by the pure-target projector test.

Displayed formulas in this passage

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$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
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$ε_{rec} (N) = D CPTP in f d_{⋄} (DN, id_{2}) .$
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$F_{e} (Λ) = ⟨ Φ ∣ (id \otimes Λ) (∣ Φ ⟩ ⟨ Φ ∣) ∣ Φ ⟩,$
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$F_{wc} (Λ) = ρ_{M} in f ⟨ ψ_{ρ} ∣ (id \otimes Λ) (∣ ψ_{ρ} ⟩ ⟨ ψ_{ρ} ∣) ∣ ψ_{ρ} ⟩ .$