Manuscript En01-A@1 · En01-A.tex · section · 558b972b

The real-positive family

Frauchiger–Renner probabilities and parameter geometry › The real-positive family · lines 2060–2098

En01-A@1#eq:fr

Designate two lab qubits $a, b$ . A preparation followed by a record-controlled rotation produces

∣Ψ ⟩ = p_{00} ∣00 ⟩ + p_{10} ∣10 ⟩ + p_{11} ∣11 ⟩, p_{00} + p_{10} + p_{11} = 1.

The record classes may be singleton. Let

∣ ok ⟩ = \frac{∣0 ⟩ - ∣1 ⟩}{2}, ∣ w ⟩ = ∣ ok ⟩ \otimes ∣ ok ⟩, M = ∣ w ⟩ ⟨ w ∣.

For a pair $π = {r, r^{'}}$ , define its scalar cross contribution

C_{M} (π) = 2 p_{r} p_{r^{'}} Re ⟨ Φ_{r^{'}} ∣ M ∣ Φ_{r} ⟩ .

Then

tr (M Δ_{J} ρ) = D_{M} + π : D (π) \cap J = \emptyset \sum C_{M} (π), D_{M} = r \sum p_{r} ⟨ Φ_{r} ∣ M ∣ Φ_{r} ⟩ .

Displayed formulas in this passage

En01-A@1#eq:fr · line 2064
$∣Ψ ⟩ = p_{00} ∣00 ⟩ + p_{10} ∣10 ⟩ + p_{11} ∣11 ⟩, p_{00} + p_{10} + p_{11} = 1.$
unlabeled · line 2077
$∣ ok ⟩ = \frac{∣0 ⟩ - ∣1 ⟩}{2}, ∣ w ⟩ = ∣ ok ⟩ \otimes ∣ ok ⟩, M = ∣ w ⟩ ⟨ w ∣.$
unlabeled · line 2085
$C_{M} (π) = 2 p_{r} p_{r^{'}} Re ⟨ Φ_{r^{'}} ∣ M ∣ Φ_{r} ⟩ .$
unlabeled · line 2091
$tr (M Δ_{J} ρ) = D_{M} + π : D (π) \cap J = \emptyset \sum C_{M} (π), D_{M} = r \sum p_{r} ⟨ Φ_{r} ∣ M ∣ Φ_{r} ⟩ .$