Manuscript Hawk02-B@2 · Hawk02-B.tex · lemma · 95ea744f

Lemma: Identity versus replacement

Conventions and elementary channel estimates · lines 272–318 · proof included or linked

Hawk02-B@2#lem:replacement

For a $d$ -dimensional system, let

c_{d} = 1 - \frac{1}{d ^{2}} .

Then

d_{⋄} (id, T_{I / d}) = c_{d}, σ in f d_{⋄} (id, T_{σ}) = c_{d} .

Proof. Temporarily index a basis by $0, \dots, d - 1$ . With $ζ = e^{2 π i / d}$ , define

X ∣ a ⟩ = ∣ a + 1 ⟩, Z ∣ a ⟩ = ζ^{a} ∣ a ⟩,

where addition is modulo $d$ . Summation on matrix units gives

T_{I / d} = \frac{1}{d ^{2}} r, s = 0 \sum^{d - 1} Ad_{X^{r} Z^{s}} .

Separating the identity term and using that the half-diamond distance between channels is at most one proves $d_{⋄} (id, T_{I / d}) \leq 1 - d^{- 2}$ .

Let

∣ Φ_{d} ⟩ = \frac{1}{d} a = 0 \sum^{d - 1} ∣ a ⟩_{C} ∣ a ⟩_{Q}, Φ_{d} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣ .

For every density operator $σ$ ,

Tr [Φ_{d} (σ \otimes I_{Q} / d)] = d^{- 2} .

The Bell-projector test distinguishes $Φ_{d}$ from the replaced output with probability difference $1 - d^{- 2}$ . It supplies the matching lower bound for every $σ$ .

Displayed formulas in this passage

unlabeled · line 275
$c_{d} = 1 - \frac{1}{d ^{2}} .$
unlabeled · line 279
$d_{⋄} (id, T_{I / d}) = c_{d}, σ in f d_{⋄} (id, T_{σ}) = c_{d} .$
unlabeled · line 289
$X ∣ a ⟩ = ∣ a + 1 ⟩, Z ∣ a ⟩ = ζ^{a} ∣ a ⟩,$
unlabeled · line 294
$T_{I / d} = \frac{1}{d ^{2}} r, s = 0 \sum^{d - 1} Ad_{X^{r} Z^{s}} .$
unlabeled · line 305
$∣ Φ_{d} ⟩ = \frac{1}{d} a = 0 \sum^{d - 1} ∣ a ⟩_{C} ∣ a ⟩_{Q}, Φ_{d} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣ .$
unlabeled · line 312
$Tr [Φ_{d} (σ \otimes I_{Q} / d)] = d^{- 2} .$

Lemma: Identity versus replacement

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