Manuscript En10-B@1 · En10-B.tex · section · a1105bad

Propagators and channel construction

Derived channel and complete acquisition instrument › Propagators and channel construction · lines 391–448

En10-B@1#eq:derived-channel, En10-B@1#eq:Enter, En10-B@1#eq:projective-target, En10-B@1#eq:acquisition

Define

V_{k} = e^{- i (ϕ + ξ_{k}) K_{n}}, U_{k} = CZ (V_{k} \otimes V_{k}^{†}) .

For $h = 1 - γ$ , put

L_{0} = ∣ 0 ⟩ ⟨ 0 ∣ + h ∣ 1 ⟩ ⟨ 1 ∣, L_{1} = γ ∣ 0 ⟩ ⟨ 1 ∣ .

The channel to be derived is

Φ_{η} (X) = k = 1 \sum^{m} ℓ = 0 \sum^{1} H_{k ℓ} X H_{k ℓ}^{†}, H_{k ℓ} = w_{k} (I \otimes L_{ℓ}) U_{k} .

Let $R = {00, 01, 10, 11}$ . To permit comparison with complete attempted-run records, use the common alphabet

R_{+} = R \cup {⊥}, H_{C} = C^{5} .

The ideal failure branch is zero. Real failure or rejection branches, if present, must remain in $R_{+}$ or in a predeclared larger alphabet.

For any finite retained factor $T$ , define

D_{r} M_{B}^{T} (X) J_{B}^{T} (X) = ⟨ r ∣_{B} \otimes I_{T}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes D_{r} X D_{r}^{†}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes Tr_{B} [(∣ r ⟩ ⟨ r ∣_{B} \otimes I_{T}) X (∣ r ⟩ ⟨ r ∣_{B} \otimes I_{T})] .

The complete acquisition map from the supplier interface is

A_{η}^{T} = M_{B}^{T} \circ (Φ_{η} \otimes id_{T}) \circ (P \otimes id_{T}) .

Displayed formulas in this passage

En10-B@1#eq:derived-channel · line 404
$Φ_{η} (X) = k = 1 \sum^{m} ℓ = 0 \sum^{1} H_{k ℓ} X H_{k ℓ}^{†}, H_{k ℓ} = w_{k} (I \otimes L_{ℓ}) U_{k} .$
En10-B@1#eq:Enter · line 426
$D_{r} M_{B}^{T} (X) J_{B}^{T} (X) = ⟨ r ∣_{B} \otimes I_{T}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes D_{r} X D_{r}^{†}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes Tr_{B} [(∣ r ⟩ ⟨ r ∣_{B} \otimes I_{T}) X (∣ r ⟩ ⟨ r ∣_{B} \otimes I_{T})] .$
En10-B@1#eq:projective-target · line 426
$D_{r} M_{B}^{T} (X) J_{B}^{T} (X) = ⟨ r ∣_{B} \otimes I_{T}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes D_{r} X D_{r}^{†}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes Tr_{B} [(∣ r ⟩ ⟨ r ∣_{B} \otimes I_{T}) X (∣ r ⟩ ⟨ r ∣_{B} \otimes I_{T})] .$
En10-B@1#eq:acquisition · line 441
$A_{η}^{T} = M_{B}^{T} \circ (Φ_{η} \otimes id_{T}) \circ (P \otimes id_{T}) .$
unlabeled · line 394
$V_{k} = e^{- i (ϕ + ξ_{k}) K_{n}}, U_{k} = CZ (V_{k} \otimes V_{k}^{†}) .$
unlabeled · line 399
$L_{0} = ∣ 0 ⟩ ⟨ 0 ∣ + h ∣ 1 ⟩ ⟨ 1 ∣, L_{1} = γ ∣ 0 ⟩ ⟨ 1 ∣ .$
unlabeled · line 417
$R_{+} = R \cup {⊥}, H_{C} = C^{5} .$