Manuscript Oai265-A@2 · Oai265-A.tex · theorem · c0a34a45

Theorem: A conservative three-correlation certificate

Finite memory certification is not entropy calibration › A separate, explicitly defined qubit procedure · lines 1676–1719 · proof included or linked

Oai265-A@2#thm:statistics

Back to manuscript

Under Specification spec:stat, with probability at least $1 - α$ , the separately compared target channel has half-diamond error from identity in

[max {0, 1 - f_{+} - b}, min {1, 2 1 - f_{-} + b}] .

Proof. Hoeffding's inequality for observations in $[- 1, 1]$ gives

P {∣ r_{s} - E r_{s} ∣ > t_{N}} \leq 2 e^{- N t_{N}^{2} /2} .

The union bound and bias allowances give $f \in [f_{-}, f_{+}]$ with the stated coverage Hoeffding.

Testing the ideal Bell projector gives

1 - f \leq d_{⋄} (Λ, id) .

For the upper bound, the normalized-Choi comparison gives

d_{⋄} (Λ, id) \leq 2 d_{tr} (ω, ∣ Ω_{2} ⟩ ⟨ Ω_{2} ∣) .

Indeed every pure referenced qubit input has the form $(B \otimes I) ∣ Ω_{2} ⟩$ with $∥ B ∥_{\infty}^{2} \leq 2$ ; sandwiching its Choi difference gives that dimension factor. The pure-target trace-distance inequality yields

d_{tr} (ω, ∣ Ω_{2} ⟩ ⟨ Ω_{2} ∣) \leq 1 - f

FuchsVanDeGraaf. Finally add the target transfer allowance by the triangle inequality.

Displayed formulas in this passage

unlabeled · line 1681
$[max {0, 1 - f_{+} - b}, min {1, 2 1 - f_{-} + b}] .$
unlabeled · line 1692
$P {∣ r_{s} - E r_{s} ∣ > t_{N}} \leq 2 e^{- N t_{N}^{2} /2} .$
unlabeled · line 1701
$1 - f \leq d_{⋄} (Λ, id) .$
unlabeled · line 1705
$d_{⋄} (Λ, id) \leq 2 d_{tr} (ω, ∣ Ω_{2} ⟩ ⟨ Ω_{2} ∣) .$
unlabeled · line 1714
$d_{tr} (ω, ∣ Ω_{2} ⟩ ⟨ Ω_{2} ∣) \leq 1 - f$

Theorem: A conservative three-correlation certificate

Displayed formulas in this passage

Nearby in this section