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

Proposition: FR profile and interface discrepancy

Frauchiger–Renner probabilities and parameter geometry › The real-positive family · lines 2099–2143 · proof included or linked

En01-A@1#prop:frprofile

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At $p_{00} = p_{10} = p_{11} = 1/3$ ,

D_{M} = \frac{1}{4}, C_{M} (00, 10) = - \frac{1}{6}, C_{M} (10, 11) = - \frac{1}{6}, C_{M} (00, 11) = \frac{1}{6} .

Consequently,

J tr (M Δ_{J} ρ) \emptyset 1/12 {1} 1/12 {2} 1/12 {1, 2} 1/4

but, on the full two-qubit interface,

δ_{{1}} = δ_{{2}} = \frac{2}{3} .

Proof. The overlaps with $00, 10, 11$ are $1/2, - 1/2, 1/2$ . Substitution gives the cross contributions and the four probabilities.

For record $1$ ,

X_{{1}} = ∣ α ⟩ ⟨ β ∣ + ∣ β ⟩ ⟨ α ∣, ∣ α ⟩ = \frac{∣00 ⟩}{3}, ∣ β ⟩ = \frac{∣10 ⟩ + ∣11 ⟩}{3} .

The vectors are orthogonal, with norm product $2 /3$ . The nonzero eigenvalues are therefore $\pm 2 /3$ . The record- $2$ calculation is the corresponding exchanged decomposition.

Displayed formulas in this passage

unlabeled · line 2102
$D_{M} = \frac{1}{4}, C_{M} (00, 10) = - \frac{1}{6}, C_{M} (10, 11) = - \frac{1}{6}, C_{M} (00, 11) = \frac{1}{6} .$
unlabeled · line 2112
$J tr (M Δ_{J} ρ) \emptyset 1/12 {1} 1/12 {2} 1/12 {1, 2} 1/4$
unlabeled · line 2120
$δ_{{1}} = δ_{{2}} = \frac{2}{3} .$
unlabeled · line 2130
$X_{{1}} = ∣ α ⟩ ⟨ β ∣ + ∣ β ⟩ ⟨ α ∣, ∣ α ⟩ = \frac{∣00 ⟩}{3}, ∣ β ⟩ = \frac{∣10 ⟩ + ∣11 ⟩}{3} .$

Proposition: FR profile and interface discrepancy

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