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

Theorem: Visibility is inaccessible-side fidelity

Cross operators, fidelity, screening, and noisy architecture · lines 1305–1338 · proof included or linked

En01-A@1#thm:fidelity, En01-A@1#eq:visibility-fidelity

For

σ^{S}_{r} = tr_{S} ∣ Φ_{r} ⟩ ⟨ Φ_{r} ∣,

one has

v_{r r^{'}} (S) = F (σ^{S}_{r}, σ^{S}_{r^{'}}) .

Proof. This is Uhlmann's variational identity Uhlmann1976. For completeness, write Schmidt decompositions

∣ Φ_{r} ⟩ = k \sum λ_{k} ∣ u_{k} ⟩ ∣ x_{k} ⟩, ∣ Φ_{r^{'}} ⟩ = l \sum μ_{l} ∣ w_{l} ⟩ ∣ y_{l} ⟩ .

The overlap optimized in Lemma lem:variational is $tr (U B)$ , where

B = k, l \sum λ_{k} μ_{l} ⟨ y_{l} ∣ x_{k} ⟩ ∣ u_{k} ⟩ ⟨ w_{l} ∣.

Its maximum is $∥ B ∥_{1}$ . In the respective orthonormal Schmidt bases, the nonzero singular values of $B$ are those of the entrywise conjugate of the matrix of $σ^{S}_{r} σ^{S}_{r^{'}}$ . Complex conjugation preserves singular values. Thus the two trace norms coincide.

Displayed formulas in this passage

En01-A@1#eq:visibility-fidelity · line 1312
$v_{r r^{'}} (S) = F (σ^{S}_{r}, σ^{S}_{r^{'}}) .$
unlabeled · line 1308
$σ^{S}_{r} = tr_{S} ∣ Φ_{r} ⟩ ⟨ Φ_{r} ∣,$
unlabeled · line 1321
$∣ Φ_{r} ⟩ = k \sum λ_{k} ∣ u_{k} ⟩ ∣ x_{k} ⟩, ∣ Φ_{r^{'}} ⟩ = l \sum μ_{l} ∣ w_{l} ⟩ ∣ y_{l} ⟩ .$
unlabeled · line 1328
$B = k, l \sum λ_{k} μ_{l} ⟨ y_{l} ∣ x_{k} ⟩ ∣ u_{k} ⟩ ⟨ w_{l} ∣.$

Theorem: Visibility is inaccessible-side fidelity

Displayed formulas in this passage

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