Manuscript Em03-B@2 · Em03-B-v2.tex · theorem · c1a0977d

Theorem: Exact assignment-noise distance

B5: a complete readout apparatus and its errors › Pointer interaction and complete instrument · lines 1498–1526 · proof included or linked

Em03-B@2#thm:assignment, Em03-B@2#eq:assignmenterror

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With the ideal output alphabet enlarged by zero-weight failure records,

d_{⋄} (P_{W}^{S}, P_{N}^{S}) = δ_{as} := j max [1 - W (y_{j} ∣ j)] .

Proof. For an arbitrary operator on the input and a reference, first pinch into the $Q$ -basis blocks $X_{j j}$ . Pinching is trace-norm contractive and $\sum_{j} ∥ X_{j j} ∥_{1} \leq ∥ X ∥_{1}$ . The difference of the two output maps has norm at most

j \sum (y \sum ∣ W (y ∣ j) - δ_{y, y_{j}} ∣) ∥ X_{j j} ∥_{1} \leq 2 δ_{as} ∥ X ∥_{1} .

This gives the complete upper bound. A basis input attaining the maximum has classical output distance $1 - W (y_{j} ∣ j)$ , proving equality.

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Em03-B@2#eq:assignmenterror · line 1502
$d_{⋄} (P_{W}^{S}, P_{N}^{S}) = δ_{as} := j max [1 - W (y_{j} ∣ j)] .$
unlabeled · line 1518
$j \sum (y \sum ∣ W (y ∣ j) - δ_{y, y_{j}} ∣) ∥ X_{j j} ∥_{1} \leq 2 δ_{as} ∥ X ∥_{1} .$

Theorem: Exact assignment-noise distance

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