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

Proposition: Non-robust single-record adequacy

Frauchiger–Renner probabilities and parameter geometry › The real-positive family · lines 2198–2234 · proof included or linked

En01-A@1#prop:frrobustness

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On the positive real simplex, friend $1$ 's dephasing is adequate for $M$ exactly when $p_{10} = p_{11}$ , and friend $2$ 's exactly when $p_{00} = p_{10}$ . Each condition is codimension one. Their simultaneous satisfaction is the symmetric point, of codimension two in the two-dimensional simplex.

For

p_{00} = \frac{1}{3}, p_{10} = \frac{1}{3} + ε, p_{11} = \frac{1}{3} - ε,

tr (M Δ_{{1}} ρ) - tr (M ρ) = \frac{ε}{2} + O (ε^{3}) .

The two-record coefficient remains nonzero for sufficiently small $ε$ , although the question order generally changes to one.

Proof. Use (eq:freta1)–(eq:freta12) and positivity. The square-root difference satisfies

\frac{1}{3} + ε - \frac{1}{3} - ε = 3 ε + O (ε^{3}) .

Multiply by $1/ (23)$ . Continuity preserves the positive two-record coefficient locally.

Displayed formulas in this passage

unlabeled · line 2208
$p_{00} = \frac{1}{3}, p_{10} = \frac{1}{3} + ε, p_{11} = \frac{1}{3} - ε,$
unlabeled · line 2213
$tr (M Δ_{{1}} ρ) - tr (M ρ) = \frac{ε}{2} + O (ε^{3}) .$
unlabeled · line 2225
$\frac{1}{3} + ε - \frac{1}{3} - ε = 3 ε + O (ε^{3}) .$

Proposition: Non-robust single-record adequacy

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