Manuscript VII-G@1 · VII-G.tex · lemma · 54f561e8

Lemma: Finite Choi certificate

Assisted stability and finite Choi certificates › Choi upper and lower bounds · lines 1605–1643 · proof included or linked

VII-G@1#lem:choi, VII-G@1#eq:choi-bounds

For every such map,

∥ χ (D) ∥_{1} \leq ∥ D ∥_{⋄} \leq d ∥ χ (D) ∥_{1} .

For a column map $D_{g}$ , its Choi matrix is the direct sum of the branch Choi matrices in the output label.

Proof. Apply $D$ to one half of the normalized maximally entangled state. Its output is $χ (D)$ , giving the lower bound.

For the upper bound, adjoin an arbitrary reference space $N$ . Any unit vector in $N \otimes C^{d}$ can be written

u = (A \otimes I) ∣ Ω ⟩, Tr (A^{†} A) = 1,

with $A : C^{d} \to N$ , and therefore $∥ A ∥_{\infty} \leq 1$ . For two unit vectors $u, v$ , represented by $A, B$ ,

(id_{N} \otimes D) (∣ u ⟩ ⟨ v ∣) = (A \otimes I) J (D) (B^{†} \otimes I) .

Its trace norm is at most $∥ J (D) ∥_{1}$ . A singular-value decomposition of an arbitrary input operator extends the bound to

∥(id_{N} \otimes D) (X) ∥_{1} \leq ∥ J (D) ∥_{1} ∥ X ∥_{1} .

Taking the supremum proves the upper bound. The direct-sum assertion follows from linearity of the Choi construction.

Displayed formulas in this passage

VII-G@1#eq:choi-bounds · line 1608
$∥ χ (D) ∥_{1} \leq ∥ D ∥_{⋄} \leq d ∥ χ (D) ∥_{1} .$
unlabeled · line 1624
$u = (A \otimes I) ∣ Ω ⟩, Tr (A^{†} A) = 1,$
unlabeled · line 1630
$(id_{N} \otimes D) (∣ u ⟩ ⟨ v ∣) = (A \otimes I) J (D) (B^{†} \otimes I) .$
unlabeled · line 1637
$∥(id_{N} \otimes D) (X) ∥_{1} \leq ∥ J (D) ∥_{1} ∥ X ∥_{1} .$