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

Theorem: Exact finite descended-evaluation pilot

A finite operational encoding of descended evaluations › Three-qubit residual and its reading instrument · lines 1858–1897 · proof included or linked

III-B@1#thm:pilot

The following hold.

Equation (eq:complete-pilot) is a complete instrument on every input, with every untouched reference.
For $σ_{k} = D_{k} ρ D_{k}^{†}$ and $p_{k} > 0$ ,
$E [B_{*} y ∣ k] = \frac{Tr ( A _{k} σ _{k} )}{p _{k}} .$
On the encoded classical input $∣ j ⟩$ ,
$E [B_{*} y ∣ k, j] = F_{k} (z_{j}) = j (k + 1) .$
The table is independent of the representative used to name each $z_{j}$ .

Proof. A recorded Kraus family is

L_{k, y} = ∣ k ⟩_{R} \otimes ∣ y ⟩_{Y} \otimes M_{k, y} D_{k} .

Its completeness is

k, y \sum L_{k, y}^{†} L_{k, y} = k \sum D_{k}^{†} I_{X} D_{k} = I_{C X} .

Tensoring with $I_{E}$ proves the complete reference statement.

Since

y = \pm 1 \sum B_{*} y E_{k, y} = A_{k},

the conditional score identity follows. The diagonal matrix entries and (eq:integer-table) give (iii). Classical Ward descent gives (iv).

Displayed formulas in this passage

unlabeled · line 1865
$E [B_{*} y ∣ k] = \frac{Tr ( A _{k} σ _{k} )}{p _{k}} .$
unlabeled · line 1870
$E [B_{*} y ∣ k, j] = F_{k} (z_{j}) = j (k + 1) .$
unlabeled · line 1880
$L_{k, y} = ∣ k ⟩_{R} \otimes ∣ y ⟩_{Y} \otimes M_{k, y} D_{k} .$
unlabeled · line 1884
$k, y \sum L_{k, y}^{†} L_{k, y} = k \sum D_{k}^{†} I_{X} D_{k} = I_{C X} .$
unlabeled · line 1891
$y = \pm 1 \sum B_{*} y E_{k, y} = A_{k},$