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

An explicit local apparatus dilation

An explicit local apparatus dilation · lines 1484–1525

En06-B@1#eq:measurementunitary, En06-B@1#eq:UL, En06-B@1#eq:dilationaction

At each wing add three blank qubits: the record carrier $R$ , an outcome environment $E$ , and a setting environment $F$ . The setting input is $S$ .

For $a = \pm 1$ , let $r (a) = (1 - a) /2 \in {0, 1}$ . For each setting $θ$ , define a unitary on $QR E$ by

C_{θ} = a = \pm 1 \sum P_{a, θ} \otimes X_{R}^{r (a)} \otimes X_{E}^{r (a)} .

It is unitary because the projectors are orthogonal and the conditional bit flips are unitary.

Let $D_{S F}$ be $CNOT_{S \to F}$ , and put

U_{L} = D_{S_{L} F_{L}} (s = 0 \sum^{1} ∣ s ⟩ ⟨ s ∣_{S_{L}} \otimes C_{θ_{s}}) \otimes I_{W},

with tensor order adjusted to the named factors. Define $U_{R}$ analogously using $φ_{t}$ . The two wing unitaries act on disjoint factor sets.

Their action on initialized apparatus factors is

∣ s ⟩_{S} ∣ ψ ⟩_{Q} ∣ 000 ⟩_{R E F} ⟼ a = \pm 1 \sum ∣ s ⟩_{S} P_{a, θ_{s}} ∣ ψ ⟩_{Q} ∣ r (a) ⟩_{R} ∣ r (a) ⟩_{E} ∣ s ⟩_{F} .

Displayed formulas in this passage

En06-B@1#eq:measurementunitary · line 1492
$C_{θ} = a = \pm 1 \sum P_{a, θ} \otimes X_{R}^{r (a)} \otimes X_{E}^{r (a)} .$
En06-B@1#eq:UL · line 1503
$U_{L} = D_{S_{L} F_{L}} (s = 0 \sum^{1} ∣ s ⟩ ⟨ s ∣_{S_{L}} \otimes C_{θ_{s}}) \otimes I_{W},$
En06-B@1#eq:dilationaction · line 1517
$∣ s ⟩_{S} ∣ ψ ⟩_{Q} ∣ 000 ⟩_{R E F} ⟼ a = \pm 1 \sum ∣ s ⟩_{S} P_{a, θ_{s}} ∣ ψ ⟩_{Q} ∣ r (a) ⟩_{R} ∣ r (a) ⟩_{E} ∣ s ⟩_{F} .$