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

Theorem: FR order two is question-level

Frauchiger–Renner probabilities and parameter geometry › The real-positive family · lines 2145–2196 · proof included or linked

En01-A@1#thm:frorder, En01-A@1#eq:freta1, En01-A@1#eq:freta2, En01-A@1#eq:freta12

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For positive real amplitudes,

η_{M} ({1}) η_{M} ({2}) η_{M} ({1, 2}) = \frac{1}{2} p_{00} (p_{10} - p_{11}), = \frac{1}{2} p_{11} (p_{10} - p_{00}), = \frac{1}{2} p_{00} p_{11} .

At the symmetric point,

ord (id) = 1, ord (M) = ord (Λ_{M}) = 2,

with weights

ω_{J} (id) ω_{J} (Λ_{M}) {1} 2 /3 0 {2} 2 /3 0 {1, 2} 1/3 1/6

The order-two assertion is about Wigner's effect or its binary measurement channel, not about the state.

Proof. Localization gives

η_{M} ({1}) = - C_{M} (00, 10) - C_{M} (00, 11),

η_{M} ({2}) = - C_{M} (00, 11) - C_{M} (10, 11), η_{M} ({1, 2}) = C_{M} (00, 11) .

Substitute the overlaps. The state singleton weights equal the discrepancies of Proposition prop:frprofile. The two-record operator $Z_{{1, 2}}$ contains only the pair $(00, 11)$ , so its half trace norm is $p_{00} p_{11}$ . The question weights follow from Corollary cor:question-order.

Displayed formulas in this passage

En01-A@1#eq:freta1 · line 2147
$η_{M} ({1}) η_{M} ({2}) η_{M} ({1, 2}) = \frac{1}{2} p_{00} (p_{10} - p_{11}), = \frac{1}{2} p_{11} (p_{10} - p_{00}), = \frac{1}{2} p_{00} p_{11} .$
En01-A@1#eq:freta2 · line 2147
$η_{M} ({1}) η_{M} ({2}) η_{M} ({1, 2}) = \frac{1}{2} p_{00} (p_{10} - p_{11}), = \frac{1}{2} p_{11} (p_{10} - p_{00}), = \frac{1}{2} p_{00} p_{11} .$
En01-A@1#eq:freta12 · line 2147
$η_{M} ({1}) η_{M} ({2}) η_{M} ({1, 2}) = \frac{1}{2} p_{00} (p_{10} - p_{11}), = \frac{1}{2} p_{11} (p_{10} - p_{00}), = \frac{1}{2} p_{00} p_{11} .$
unlabeled · line 2161
$ord (id) = 1, ord (M) = ord (Λ_{M}) = 2,$
unlabeled · line 2167
$ω_{J} (id) ω_{J} (Λ_{M}) {1} 2 /3 0 {2} 2 /3 0 {1, 2} 1/3 1/6$
unlabeled · line 2181
$η_{M} ({1}) = - C_{M} (00, 10) - C_{M} (00, 11),$
unlabeled · line 2184
$η_{M} ({2}) = - C_{M} (00, 11) - C_{M} (10, 11), η_{M} ({1, 2}) = C_{M} (00, 11) .$

Theorem: FR order two is question-level

Displayed formulas in this passage

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