Manuscript Psy02-B@2 · Psy02-B-v2.tex · section · cfd06e2e

Code and mass assignments

Track A1: carrier, matched branches, and exact dynamics › Code and mass assignments · lines 259–301

Psy02-B@2#eq:massbranches

Let

H_{C} = C ∣ Ω ⟩ \oplus span {∣ 0 ⟩, ∣ 1 ⟩},

and define

P = ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣, Z = ∣ 0 ⟩ ⟨ 0 ∣ - ∣ 1 ⟩ ⟨ 1 ∣, X = ∣ 0 ⟩ ⟨ 1 ∣ + ∣ 1 ⟩ ⟨ 0 ∣ .

They vanish on $Ω$ . The level $Ω$ denotes loss from the prepared code, not disappearance of matter.

Successful preparation is

ρ_{+} = ∣ + ⟩ ⟨ + ∣, ∣ + ⟩ = \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2} .

The comparison preparation is

Δ ρ_{+} = \frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣) .

Preparation succeeds with probability $p$ ; failure is recorded.

For centers $R_{i} \in R^{3}$ , masses $m_{i} > 0$ , displacement $u \in R^{3}$ , pattern $s_{i} \in {- 1, 0, 1}$ , and $a > 0$ , put

g_{a} (x) μ_{0} (x) μ_{1} (x) = (2 π a^{2})^{- 3/2} e^{- \abs x^{2} / (2 a^{2})}, = μ_{rest} (x) + i \sum m_{i} g_{a} (x - R_{i} - s_{i} u /2), = μ_{rest} (x) + i \sum m_{i} g_{a} (x - R_{i} + s_{i} u /2) .

Use $δ μ = μ_{0} - μ_{1}$ and $μ_{Ω} = (μ_{0} + μ_{1}) /2$ . The common rest density cancels.

Displayed formulas in this passage

Psy02-B@2#eq:massbranches · line 290
$g_{a} (x) μ_{0} (x) μ_{1} (x) = (2 π a^{2})^{- 3/2} e^{- \abs x^{2} / (2 a^{2})}, = μ_{rest} (x) + i \sum m_{i} g_{a} (x - R_{i} - s_{i} u /2), = μ_{rest} (x) + i \sum m_{i} g_{a} (x - R_{i} + s_{i} u /2) .$
unlabeled · line 262
$H_{C} = C ∣ Ω ⟩ \oplus span {∣ 0 ⟩, ∣ 1 ⟩},$
unlabeled · line 268
$P = ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣, Z = ∣ 0 ⟩ ⟨ 0 ∣ - ∣ 1 ⟩ ⟨ 1 ∣, X = ∣ 0 ⟩ ⟨ 1 ∣ + ∣ 1 ⟩ ⟨ 0 ∣ .$
unlabeled · line 278
$ρ_{+} = ∣ + ⟩ ⟨ + ∣, ∣ + ⟩ = \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2} .$
unlabeled · line 283
$Δ ρ_{+} = \frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣) .$