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

Graph and supplier input

The frozen finite source and generator model › Graph and supplier input · lines 231–275

En10-B@1#eq:source-input, En10-B@1#eq:preparation, En10-B@1#eq:source-B

Freeze the graph

G_{2} = ({b, d}, {{b, d}}),

with basis $0 = b$ , $1 = d$ . The labels may be read as bulge and disk in the analogy, but carry no Galactic radius or metric.

Let

H_{J} = C^{2}, H_{N} = H_{A} = C^{2}, H_{B} = H_{N} \otimes H_{A} .

The supplier delivers a source qubit $J$ in

τ_{θ} = θ ∣ 0 ⟩ ⟨ 0 ∣ + (1 - θ) ∣ 1 ⟩ ⟨ 1 ∣, 0 \leq θ \leq 1.

Its source observable is $F_{J} = ∣ 0 ⟩ ⟨ 0 ∣$ , so that $Tr (F_{J} τ_{θ}) = θ$ .

Define the fixed preparation isometry

S : H_{J} ⟶ H_{B}, S ∣ j ⟩ = ∣ j j ⟩, P (X) = S X S^{†} .

Operationally, the supplied name becomes $N$ , a fresh qubit $A$ is allocated in $0$ , and $CNOT_{N \to A}$ is applied. Consequently

P (τ_{θ}) = ρ_{θ} = θ ∣ 00 ⟩ ⟨ 00 ∣ + (1 - θ) ∣ 11 ⟩ ⟨ 11 ∣ .

The map $P$ is independent of $θ$ . It does not clone arbitrary source states: a coherent input generally produces an entangled pair.

Displayed formulas in this passage

En10-B@1#eq:source-input · line 247
$τ_{θ} = θ ∣ 0 ⟩ ⟨ 0 ∣ + (1 - θ) ∣ 1 ⟩ ⟨ 1 ∣, 0 \leq θ \leq 1.$
En10-B@1#eq:preparation · line 256
$S : H_{J} ⟶ H_{B}, S ∣ j ⟩ = ∣ j j ⟩, P (X) = S X S^{†} .$
En10-B@1#eq:source-B · line 266
$P (τ_{θ}) = ρ_{θ} = θ ∣ 00 ⟩ ⟨ 00 ∣ + (1 - θ) ∣ 11 ⟩ ⟨ 11 ∣ .$
unlabeled · line 234
$G_{2} = ({b, d}, {{b, d}}),$
unlabeled · line 241
$H_{J} = C^{2}, H_{N} = H_{A} = C^{2}, H_{B} = H_{N} \otimes H_{A} .$