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

Remark: What must actually be bounded

Referenced certification interfaces › Normalized Choi and matrix-unit conversions · lines 1873–1898

The Choi estimate requires the full referenced state, not its accessible marginal. Matrix-unit estimates include off-diagonal units, which are not density matrices. Bounds for only $E_{00}$ and $E_{11}$ do not certify coherence or diamond distance.

One may reconstruct the off-diagonal unit from physical preparations:

E_{01} = ρ_{+} + i ρ_{+ i} - \frac{1 + i}{2} (E_{00} + E_{11}),

where

ρ_{+} = ∣ + ⟩ ⟨ + ∣, ρ_{+ i} = \frac{1}{2} (∣0 ⟩ + i ∣1 ⟩) (⟨ 0∣ - i ⟨ 1∣) .

Linearity and the triangle inequality then convert certified preparation-output bounds into a matrix-unit bound.

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unlabeled · line 1882
$E_{01} = ρ_{+} + i ρ_{+ i} - \frac{1 + i}{2} (E_{00} + E_{11}),$
unlabeled · line 1889
$ρ_{+} = ∣ + ⟩ ⟨ + ∣, ρ_{+ i} = \frac{1}{2} (∣0 ⟩ + i ∣1 ⟩) (⟨ 0∣ - i ⟨ 1∣) .$

Remark: What must actually be bounded

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