Manuscript Oai280-B@1 · Oai280-B.tex · lemma · 6d92ab75

Lemma: Bell-overlap bounds

Bell correlations and simultaneous finite-data certification › Choi overlap and reference-preserving conversion · lines 1254–1299 · proof included or linked

Oai280-B@1#lem:bellbounds, Oai280-B@1#eq:choiconversion

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For a channel with input and output dimension $d$ , normalized Bell Choi state $J_{Λ}$ , and Bell overlap $F$ ,

1 - F \leq d_{⋄} (Λ, id_{d}) \leq min {1, d 1 - F} .

More generally, for channels with common input dimension $D$ ,

d_{⋄} (Φ, Ψ) \leq min {1, D d_{tr} (J_{Φ}, J_{Ψ})} .

Proof. The Bell input and Bell-projector effect give the lower bound. For a state $ρ$ and pure projector $P$ ,

d_{tr} (ρ, P) \leq 1 - Tr (P ρ) .

This follows from the pure-state distance formula, convex decomposition of $ρ$ , and concavity of the square root; it is also the pure-target case of FuchsVanDeGraaf.

To prove the Choi conversion, let $Δ = Φ - Ψ$ . Unit referenced vectors can be written

u = (A \otimes I) Φ_{D}, v = (B \otimes I) Φ_{D},

where $A, B$ map the $D$ -dimensional reference into the actual reference space and have Hilbert–Schmidt norm $D$ . Then

(id \otimes Δ) (∣ u ⟩ ⟨ v ∣) = (A \otimes I) J_{Δ} (B^{†} \otimes I)

has trace norm at most $D ∥ J_{Δ} ∥_{1}$ . A singular-value decomposition extends this bound to every trace-norm-one referenced operator. Taking the supremum and dividing by two proves (eq:choiconversion). The pure-target state bound gives the identity-channel result.

Displayed formulas in this passage

Oai280-B@1#eq:choiconversion · line 1263
$d_{⋄} (Φ, Ψ) \leq min {1, D d_{tr} (J_{Φ}, J_{Ψ})} .$
unlabeled · line 1258
$1 - F \leq d_{⋄} (Λ, id_{d}) \leq min {1, d 1 - F} .$
unlabeled · line 1273
$d_{tr} (ρ, P) \leq 1 - Tr (P ρ) .$
unlabeled · line 1283
$u = (A \otimes I) Φ_{D}, v = (B \otimes I) Φ_{D},$
unlabeled · line 1290
$(id \otimes Δ) (∣ u ⟩ ⟨ v ∣) = (A \otimes I) J_{Δ} (B^{†} \otimes I)$

Nearby in this section

section: Choi overlap and reference-preserving conversion