Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

Three-qubit residual and its reading instrument

A finite operational encoding of descended evaluations › Three-qubit residual and its reading instrument · lines 1822–1857

III-B@1#eq:complete-pilot

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Let

H_{X} = (C^{2})^{\otimes 3},

with computational basis $∣ j ⟩$ , $0 \leq j \leq 7$ . Declare the encoding $z_{j} \mapsto ∣ j ⟩$ , and set

A_{k} = j = 0 \sum^{7} j (k + 1) ∣ j ⟩ ⟨ j ∣, B_{*} = 28.

The largest eigenvalue is $28$ .

Define effects and Kraus operators

E_{k, y} = \frac{1}{2} (I + \frac{y A _{k}}{B _{*}}), M_{k, y} = E_{k, y}^{1/2}, y \in {+ 1, - 1} .

These are positive and $\sum_{y} M_{k, y}^{†} M_{k, y} = I$ .

After receiving the classical clock code, read this binary instrument and retain the classical outcome $y$ . The complete clock–evaluation instrument is

I (Z) = k = 0 \sum^{3} y = \pm 1 \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes M_{k, y} D_{k} Z D_{k}^{†} M_{k, y}^{†} .

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III-B@1#eq:complete-pilot · line 1849
$I (Z) = k = 0 \sum^{3} y = \pm 1 \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes M_{k, y} D_{k} Z D_{k}^{†} M_{k, y}^{†} .$
unlabeled · line 1825
$H_{X} = (C^{2})^{\otimes 3},$
unlabeled · line 1830
$A_{k} = j = 0 \sum^{7} j (k + 1) ∣ j ⟩ ⟨ j ∣, B_{*} = 28.$
unlabeled · line 1837
$E_{k, y} = \frac{1}{2} (I + \frac{y A _{k}}{B _{*}}), M_{k, y} = E_{k, y}^{1/2}, y \in {+ 1, - 1} .$

Three-qubit residual and its reading instrument

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