Manuscript Oai069-A@1 · Oai069-A.tex · proposition · a4913aab

Proposition: Complete finite coverage

Complete coverage and simultaneous finite-confidence certification › All matrix units, not a few selected observables · lines 1012–1033 · proof included or linked

Oai069-A@1#prop:channelcoverage

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Values of $Φ_{ν, k}$ on every matrix unit of $B (C \otimes K)$ determine the entire block channel. Testing every sector determines the complete channel on $A_{in}$ , including every reference and every gauge correlation.

Equivalently, its normalized Choi state determines the channel.

Proof. Matrix units span the input operator space. Their coefficients are precisely the blocks of

J (Φ_{ν, k}) = \frac{1}{n} i, j \sum ∣ i ⟩ ⟨ j ∣ \otimes Φ_{ν, k} (∣ i ⟩ ⟨ j ∣) .

Thus either set of data gives the other and determines the linear map. Direct-sum decomposition and reference extension give the stated complete scope.

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unlabeled · line 1026
$J (Φ_{ν, k}) = \frac{1}{n} i, j \sum ∣ i ⟩ ⟨ j ∣ \otimes Φ_{ν, k} (∣ i ⟩ ⟨ j ∣) .$

Proposition: Complete finite coverage

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