Manuscript En02-A@1 · En02-A-v1.tex · theorem · bb1319d5

Theorem: Full-capacity outcome rigidity under global faithfulness

Diagonal-faithful rigidity › Global diagonal faithfulness removes the boundary · lines 1609–1649 · proof included or linked

En02-A@1#thm:boundary

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Let $N \geq 3$ , $S \in A_{N}$ , and assume $DF_{\leq N}$ . Then every outcome law on every input with $n \leq N$ is Born.

Proof. Theorem thm:I handles $n < N$ . Take $n = N$ and first suppose every computational amplitude of $ψ$ is nonzero. Partition its names into nonempty $A, B$ . Every first-readout residual has fewer than $N$ qubits and therefore has Born laws by Theorem thm:I. Global diagonal faithfulness identifies its computational diagonal with the original input's projective conditional diagonal. Consequently,

P^{L}_{ψ} (κ, λ) = P^{A}_{ψ} (κ) \frac{b ^{L} _{ψ} ( κ , λ )}{b ^{A} _{ψ} ( κ )} = P^{B}_{ψ} (λ) \frac{b ^{L} _{ψ} ( κ , λ )}{b ^{B} _{ψ} ( λ )} .

The formula remains true when a strategy first weight is zero. All Born weights in the display are positive, so

\frac{P ^{A} _{ψ} ( κ )}{b ^{A} _{ψ} ( κ )} = \frac{P ^{B} _{ψ} ( λ )}{b ^{B} _{ψ} ( λ )}

for every pair. These ratios have one common value. Normalization makes it one, and hence the full law is Born. Composability gives every measured marginal.

Full-support rays are dense: perturb a vector in a direction with all coordinates nonzero, avoid the finitely many perturbation values that cancel a coordinate, and normalize. Continuity of the strategy laws and Born laws extends the conclusion to every ray.

Displayed formulas in this passage

unlabeled · line 1624
$P^{L}_{ψ} (κ, λ) = P^{A}_{ψ} (κ) \frac{b ^{L} _{ψ} ( κ , λ )}{b ^{A} _{ψ} ( κ )} = P^{B}_{ψ} (λ) \frac{b ^{L} _{ψ} ( κ , λ )}{b ^{B} _{ψ} ( λ )} .$
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$\frac{P ^{A} _{ψ} ( κ )}{b ^{A} _{ψ} ( κ )} = \frac{P ^{B} _{ψ} ( λ )}{b ^{B} _{ψ} ( λ )}$

Theorem: Full-capacity outcome rigidity under global faithfulness

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