Manuscript En06-B@1 · En06-B.tex · section · ee886ae8

The complete channel

A nontrivial finite quantum instrument › The complete channel · lines 1327–1362

En06-B@1#eq:BellKraus, En06-B@1#eq:Bellchannel

The input space is

H_{in} = Q_{L} \otimes Q_{R} \otimes W \otimes S_{L} \otimes S_{R} .

For a matrix $ρ$ on this space, let

ρ_{s t} = ⟨ s t ∣_{S_{L} S_{R}} ρ ∣ s t ⟩_{S_{L} S_{R}} .

Define

K_{ab ∣ s t} = (P_{a, θ_{s}} \otimes P_{b, φ_{t}} \otimes I_{W}) (U_{B} \otimes I_{W}) .

Using the label $a = \pm 1$ for a classical basis vector, set

Φ (ρ) = s, t \in {0, 1} a, b \in {\pm 1} \sum K_{ab ∣ s t} ρ_{s t} K_{ab ∣ s t}^{†} \otimes ∣ s t ⟩ ⟨ s t ∣_{S_{L} S_{R}} \otimes ∣ a, b ⟩ ⟨ a, b ∣_{R_{L} R_{R}} .

This specifies the canonical full-matrix extension which dephases the setting inputs. On the declared classical input algebra, it is the controlled local instrument.

Displayed formulas in this passage

unlabeled · line 1330
$H_{in} = Q_{L} \otimes Q_{R} \otimes W \otimes S_{L} \otimes S_{R} .$
unlabeled · line 1336
$ρ_{s t} = ⟨ s t ∣_{S_{L} S_{R}} ρ ∣ s t ⟩_{S_{L} S_{R}} .$
En06-B@1#eq:BellKraus · line 1340
$K_{ab ∣ s t} = (P_{a, θ_{s}} \otimes P_{b, φ_{t}} \otimes I_{W}) (U_{B} \otimes I_{W}) .$
En06-B@1#eq:Bellchannel · line 1348
$Φ (ρ) = s, t \in {0, 1} a, b \in {\pm 1} \sum K_{ab ∣ s t} ρ_{s t} K_{ab ∣ s t}^{†} \otimes ∣ s t ⟩ ⟨ s t ∣_{S_{L} S_{R}} \otimes ∣ a, b ⟩ ⟨ a, b ∣_{R_{L} R_{R}} .$

The complete channel

Displayed formulas in this passage

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