Manuscript Hawk02-C@2 · Hawk02-C.tex · lemma · bb332d66

Lemma: Identity versus replacement

Conventions and common-space comparison · lines 415–447 · proof included or linked

Hawk02-C@2#lem:replacement

On $C^{d}$ ,

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

Proof. Let $X$ be the cyclic shift and $Z$ the phase operator in a basis indexed 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 proves the upper bound $1 - d^{- 2}$ .

Let

∣ Φ_{d} ⟩ = d^{- 1/2} a \sum ∣ a ⟩_{C} ∣ a ⟩_{Q}, Φ_{d} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣ .

For every state $σ$ ,

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

The Bell-projector test supplies the matching lower bound for every replacement state.

Displayed formulas in this passage

unlabeled · line 418
$d_{⋄} (id, T_{I / d}) = c_{d}, σ in f d_{⋄} (id, T_{σ}) = c_{d} .$
unlabeled · line 428
$T_{I / d} = \frac{1}{d ^{2}} r, s = 0 \sum^{d - 1} Ad_{X^{r} Z^{s}} .$
unlabeled · line 437
$∣ Φ_{d} ⟩ = d^{- 1/2} a \sum ∣ a ⟩_{C} ∣ a ⟩_{Q}, Φ_{d} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣ .$
unlabeled · line 442
$Tr [Φ_{d} (σ \otimes I_{Q} / d)] = d^{- 2} .$