Manuscript Hawk01-B@1 · Hawk01-B.tex · proposition · cfa976f1

Proposition: Bell tests and dimension-dependent conversion

Exact and approximate residual-leakage certificates › Single-reference diagnostics · lines 486–518 · proof included or linked

Hawk01-B@1#prop:Bell, Hawk01-B@1#eq:Bellconversion, Hawk01-B@1#eq:Choiconversion

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For every $Λ : A \to A$ ,

1 - F_{e} (Λ) \leq D_{⋄} (Λ, id_{A}) \leq min {1, d 1 - F_{e} (Λ)} .

For any two channels with input dimension $d$ ,

D_{tr} (J (Φ), J (Ψ)) \leq D_{⋄} (Φ, Ψ) \leq d D_{tr} (J (Φ), J (Ψ)) .

Proof. The Bell input and Bell-projector effect give the lower bound in (eq:Bellconversion). The Choi trace-norm comparison gives (eq:Choiconversion); equivalently, the raw diamond norm is at most the trace norm of the unnormalized Choi matrix, which is $dJ$ Watrous2018. Finally, the trace distance to a pure state is at most the square root of one minus its overlap. Combine these statements.

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Hawk01-B@1#eq:Bellconversion · line 489
$1 - F_{e} (Λ) \leq D_{⋄} (Λ, id_{A}) \leq min {1, d 1 - F_{e} (Λ)} .$
Hawk01-B@1#eq:Choiconversion · line 498
$D_{tr} (J (Φ), J (Ψ)) \leq D_{⋄} (Φ, Ψ) \leq d D_{tr} (J (Φ), J (Ψ)) .$

Proposition: Bell tests and dimension-dependent conversion

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