Manuscript Hawk07-B@1 · Hawk07-B.tex · theorem · b9785538

Theorem: An exact hybrid function: teleportation

Hybrid communication and causal limitations · lines 1479–1512 · proof included or linked

Hawk07-B@1#thm:teleport

For $d = 2^{ℓ}$ , prepare $ℓ$ Bell pairs $H E$ before the unknown payload. A Bell transform on $L H$ , followed by Enter, implements

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

where $W_{z}$ are Pauli corrections. The receiver

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

satisfies $D M = id_{L}$ , including every reference. Before $Y$ arrives, the receiver's channel is

X ⟼ Tr (X) I_{E} / d .

The communicated label has $2 ℓ$ classical bits.

Proof. The coherent Bell expansion is

∣ ψ ⟩ ⟼ \frac{1}{d} z \sum ∣ z ⟩ W_{z} ∣ ψ ⟩ .

Destructive readout gives the displayed hybrid channel. Conditional correction returns every matrix unit. Forgetting the label averages the $d^{2}$ Pauli conjugations, which is depolarization.

Displayed formulas in this passage

unlabeled · line 1483
$M (X) = \frac{1}{d ^{2}} z = 1 \sum^{d^{2}} ∣ z ⟩ ⟨ z ∣_{Y} \otimes W_{z} X W_{z}^{†},$
unlabeled · line 1490
$D (Z) = z \sum W_{z}^{†} (⟨ z ∣_{Y} \otimes I_{E}) Z (∣ z ⟩_{Y} \otimes I_{E}) W_{z}$
unlabeled · line 1496
$X ⟼ Tr (X) I_{E} / d .$
unlabeled · line 1504
$∣ ψ ⟩ ⟼ \frac{1}{d} z \sum ∣ z ⟩ W_{z} ∣ ψ ⟩ .$

Theorem: An exact hybrid function: teleportation

Displayed formulas in this passage

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