Manuscript Hawk04-A@1 · Hawk04-A.tex · lemma · 526092ba

Lemma: Choi and matrix-unit bounds

Quantitative channel certificates › Normalization checks · lines 781–825 · proof included or linked

Hawk04-A@1#lem:choi

For channels $Φ, Ψ$ with input dimension $d$ ,

d_{⋄} (Φ, Ψ) \leq min {1, \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1}} .

∥ (Φ - Ψ) (E_{ij}) ∥_{1} \leq r for every i, j,

then

d_{⋄} (Φ, Ψ) \leq min {1, d r /2} .

Proof. A pure referenced input can be written as

∣ ψ ⟩ = (L \otimes I_{C}) ∣ Φ_{d} ⟩, Tr (L^{†} L) = d, ∥ L ∥^{2} \leq d .

Multiplication on both sides of the Choi difference therefore bounds its trace norm by $d$ times the Choi trace norm.

Alternatively, write

∣ ψ ⟩ = i \sum ∣ v_{i} ⟩_{Q} ∣ i ⟩_{C}, i \sum ∥ v_{i} ∥^{2} = 1.

Its output difference has trace norm at most

r i, j \sum ∥ v_{i} ∥ ∥ v_{j} ∥ \leq d r .

Divide by two, use convexity, and take the input supremum.

Displayed formulas in this passage

unlabeled · line 784
$d_{⋄} (Φ, Ψ) \leq min {1, \frac{d}{2} ∥ J_{n} (Φ) - J_{n} (Ψ) ∥_{1}} .$
unlabeled · line 792
$∥ (Φ - Ψ) (E_{ij}) ∥_{1} \leq r for every i, j,$
unlabeled · line 797
$d_{⋄} (Φ, Ψ) \leq min {1, d r /2} .$
unlabeled · line 804
$∣ ψ ⟩ = (L \otimes I_{C}) ∣ Φ_{d} ⟩, Tr (L^{†} L) = d, ∥ L ∥^{2} \leq d .$
unlabeled · line 814
$∣ ψ ⟩ = i \sum ∣ v_{i} ⟩_{Q} ∣ i ⟩_{C}, i \sum ∥ v_{i} ∥^{2} = 1.$
unlabeled · line 820
$r i, j \sum ∥ v_{i} ∥ ∥ v_{j} ∥ \leq d r .$