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

Theorem: Positive deformation at every finite size

The independent static marginal problem › An all-size positive non-Born family · lines 2889–2965 · proof included or linked

I@2#thm:static-positive

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There exist normalized, positive, continuous, marginally compatible, diagonally anchored, permutation-equivariant static families different from the Born diagonal assignment.

Proof. Fix $0 < η < 1$ . For a one-qubit state $τ$ , define

θ (τ) = η Re τ_{01}, W (τ) = e^{- i θ (τ) Y}, Y = (0 i - i 0) .

For $ρ$ on $S$ , set

W_{S} (ρ) = i \in S ⨂ W (ρ_{i}), F_{S} (ρ)_{x} = ⟨ x ∣ W_{S} (ρ) ρ W_{S} (ρ)^{†} ∣ x ⟩ .

Use the unique normalized assignment for the empty set.

For fixed $ρ$ , this is the diagonal of a density operator, so it is positive and normalized. Partial trace, matrix exponential, and diagonal extraction are continuous. If $ρ$ is diagonal, each local angle is zero, so anchoring holds. Using one identical rule at all sites gives permutation equivariance.

Define

E_{i, x_{i}} (ρ_{i}) = W (ρ_{i})^{†} ∣ x_{i} ⟩ ⟨ x_{i} ∣ W (ρ_{i}) .

Then

F_{S} (ρ)_{x} = tr (ρ i \in S ⨂ E_{i, x_{i}} (ρ_{i})) .

Summing over $S ∖ T$ gives

x_{S ∖ T} \sum F_{S} (ρ)_{x} = tr (ρ [i \in T ⨂ E_{i, x_{i}} (ρ_{i}) \otimes I_{S ∖ T}]) = tr (ρ_{T} i \in T ⨂ E_{i, x_{i}} ((ρ_{T})_{i})) = F_{T} (ρ_{T})_{x_{T}},

using local completeness and $(ρ_{T})_{i} = ρ_{i}$ .

Finally, for $ρ = ∣ + ⟩ ⟨ + ∣$ , the angle is $η /2$ . Since

e^{- i θ Y} = (cos θ sin θ - sin θ cos θ),

F_{{1}} (∣ + ⟩ ⟨ + ∣)_{0} = \frac{1 - sin η}{2} \neq = \frac{1}{2} .

The family is non-Born.

Displayed formulas in this passage

unlabeled · line 2899
$θ (τ) = η Re τ_{01}, W (τ) = e^{- i θ (τ) Y}, Y = (0 i - i 0) .$
unlabeled · line 2907
$W_{S} (ρ) = i \in S ⨂ W (ρ_{i}), F_{S} (ρ)_{x} = ⟨ x ∣ W_{S} (ρ) ρ W_{S} (ρ)^{†} ∣ x ⟩ .$
unlabeled · line 2923
$E_{i, x_{i}} (ρ_{i}) = W (ρ_{i})^{†} ∣ x_{i} ⟩ ⟨ x_{i} ∣ W (ρ_{i}) .$
unlabeled · line 2928
$F_{S} (ρ)_{x} = tr (ρ i \in S ⨂ E_{i, x_{i}} (ρ_{i})) .$
unlabeled · line 2933
$x_{S ∖ T} \sum F_{S} (ρ)_{x} = tr (ρ [i \in T ⨂ E_{i, x_{i}} (ρ_{i}) \otimes I_{S ∖ T}]) = tr (ρ_{T} i \in T ⨂ E_{i, x_{i}} ((ρ_{T})_{i})) = F_{T} (ρ_{T})_{x_{T}},$
unlabeled · line 2952
$e^{- i θ Y} = (cos θ sin θ - sin θ cos θ),$
unlabeled · line 2960
$F_{{1}} (∣ + ⟩ ⟨ + ∣)_{0} = \frac{1 - sin η}{2} \neq = \frac{1}{2} .$

Nearby in this section

section: An all-size positive non-Born family