Manuscript I@2 · I.tex · theorem · 85631fa7

Theorem: Uniform correlations

Locality and uniform correlations · lines 1461–1509 · proof included or linked

I@2#thm:uniformity

Assume $(C)^{⋆}, (NS)^{⋆}, (N)^{⋆}$ . If

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

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

Proof. A diagonal environment unitary removes the phases without changing the $S$ -law. Write

p = P^{S}_{ψ}, q = P^{B}_{ψ}, ν = P^{S \cup B}_{ψ} .

Nullity restricts $ν$ to $(k, η_{k})$ . Composability gives

p (k) = ν (k, η_{k}) = q (η_{k}), k \in K \sum p (k) = 1.

For a permutation $π$ of $K$ , extend it to a computational-basis permutation unitary $Π_{π}$ on $S$ . Then

Π_{π} ψ = \frac{1}{d} m \in K \sum ∣ m ⟩ ∣ η_{π^{- 1} (m)} ⟩ .

The same vector is $W ψ$ , where $W$ is an environment unitary sending $η_{k}$ to $η_{π^{- 1} (k)}$ . Its $S$ -law remains $p$ , by invariance under $W$ . Its $B$ -law remains $q$ , by invariance under $Π_{π}$ .

For the new pairing, nullity and composability imply

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

Every permutation preserves $p$ , so $p (k) = 1/ d$ .

Displayed formulas in this passage

unlabeled · line 1465
$ψ = \frac{1}{d} k \in K \sum e^{i θ_{k}} ∣ k ⟩_{S} ∣ η_{k} ⟩_{B}, d = ∣ K ∣,$
unlabeled · line 1479
$p = P^{S}_{ψ}, q = P^{B}_{ψ}, ν = P^{S \cup B}_{ψ} .$
unlabeled · line 1485
$p (k) = ν (k, η_{k}) = q (η_{k}), k \in K \sum p (k) = 1.$
unlabeled · line 1494
$Π_{π} ψ = \frac{1}{d} m \in K \sum ∣ m ⟩ ∣ η_{π^{- 1} (m)} ⟩ .$
unlabeled · line 1505
$p (m) = q (η_{π^{- 1} (m)}) = p (π^{- 1} (m)) .$

Theorem: Uniform correlations

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