Manuscript Hawk07-A@1 · Hawk07-A.tex · example · f9c0f5da

Example: Enter plus surviving quantum memory: teleportation

The classical obstruction and the surviving quantum mouth · lines 1494–1535

Hawk07-A@1#ex:teleport

Take $d = 2^{ℓ}$ and prepare $ℓ$ Bell pairs $H E$ before the unknown diary is supplied. A Bell transform on $L H$ gives the complete expansion

V_{tel} ∣ ψ ⟩ = \frac{1}{d} z = 1 \sum^{d^{2}} ∣ z ⟩_{T} W_{z} ∣ ψ ⟩_{E} \otimes ∣ 0 ⟩_{S},

where $W_{z}$ are fixed Pauli corrections and $S$ is an untouched two-dimensional spectator.

Enter on $T$ returns the hybrid channel

M (X) = \frac{1}{d ^{2}} z \sum ∣ z ⟩ ⟨ z ∣_{Y} \otimes W_{z} X W_{z}^{†} .

The CPTP receiver

D_{tel} (Z) = z \sum W_{z}^{†} (⟨ z ∣_{Y} \otimes I_{E}) Z (∣ z ⟩_{Y} \otimes I_{E}) W_{z}

satisfies $D_{tel} M = id_{L}$ , including references.

A complete measurement realization is the isometry

∣ z ⟩_{T} ⟼ ∣ z ⟩_{Y} ∣ z ⟩_{G} .

For the resulting complete channel $L \to (Y E) \otimes (GS)$ , the complement is

T_{I_{G} / d^{2} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{S}} .

Thus its physical residual is positive dimensional, although the returned classical record alone is input independent.

Displayed formulas in this passage

unlabeled · line 1499
$V_{tel} ∣ ψ ⟩ = \frac{1}{d} z = 1 \sum^{d^{2}} ∣ z ⟩_{T} W_{z} ∣ ψ ⟩_{E} \otimes ∣ 0 ⟩_{S},$
unlabeled · line 1509
$M (X) = \frac{1}{d ^{2}} z \sum ∣ z ⟩ ⟨ z ∣_{Y} \otimes W_{z} X W_{z}^{†} .$
unlabeled · line 1516
$D_{tel} (Z) = z \sum W_{z}^{†} (⟨ z ∣_{Y} \otimes I_{E}) Z (∣ z ⟩_{Y} \otimes I_{E}) W_{z}$
unlabeled · line 1525
$∣ z ⟩_{T} ⟼ ∣ z ⟩_{Y} ∣ z ⟩_{G} .$
unlabeled · line 1530
$T_{I_{G} / d^{2} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{S}} .$