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

Lemma: Uniform correlations inside the budget

Finite operational strategies and explicit requirements › Uniform correlations without an extra outcome-covariance axiom · lines 564–596 · proof included or linked

En02-A@1#lem:uniformity

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Suppose

ψ = \frac{1}{d} k \in K \sum e^{i θ_{k}} ∣ k ⟩_{S} ∣ η_{k} ⟩_{B}

fits capacity $N$ , where the strings $k$ are distinct and the computational strings $η_{k}$ are distinct. Then $P^{S}_{ψ}$ is uniform on $K$ .

Proof. A diagonal unitary on $B$ removes the phases without changing the $S$ -law. Write $p = P^{S}_{ψ}$ , $q = P^{B}_{ψ}$ , and $ν = P^{S \cup B}_{ψ}$ . Nullity restricts $ν$ to $(k, η_{k})$ ; composability gives

p (k) = ν (k, η_{k}) = q (η_{k}) .

For a permutation $π$ of $K$ , apply the basis permutation on $S$ sending $k$ to $π (k)$ . The resulting vector can also be obtained by a unitary on $B$ sending $η_{k}$ to $η_{π^{- 1} (k)}$ . Remote invariance therefore leaves the $S$ -law equal to $p$ and the $B$ -law equal to $q$ . Nullity for the new pairing gives

p (m) = q (η_{π^{- 1} (m)}) = p (π^{- 1} (m)) .

Thus every permutation preserves $p$ , and normalization makes it uniform. No extra qubits are allocated.

Displayed formulas in this passage

unlabeled · line 567
$ψ = \frac{1}{d} k \in K \sum e^{i θ_{k}} ∣ k ⟩_{S} ∣ η_{k} ⟩_{B}$
unlabeled · line 582
$p (k) = ν (k, η_{k}) = q (η_{k}) .$
unlabeled · line 591
$p (m) = q (η_{π^{- 1} (m)}) = p (π^{- 1} (m)) .$

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section: Uniform correlations without an extra outcome-covariance axiom