Manuscript En02-C@1 · En02-C-v1.tex · section · 7549c6df

Coherent projections

C5: amplification and the single-copy ledger › Coherent projections · lines 1227–1260

Before destructive readout, define the isometry

T : E D ⟶ D C^{'} D^{'} M, T = (I_{D} \otimes U^{*}) (∣ Φ_{m} ⟩_{F M} \otimes I_{E D}) .

Let

P = T T^{†}, Q = ∣ Φ_{δ} ⟩ ⟨ Φ_{δ} ∣_{D D^{'}} \otimes I_{C^{'} M} .

Then

T^{†} QT = L^{†} L = \frac{1}{m ^{2}} S .

The reflection about $P$ is executable as

I - 2 P = (I_{D} \otimes U^{*}) [I - 2 (∣ Φ_{m} ⟩ ⟨ Φ_{m} ∣_{F M} \otimes I_{D E})] (I_{D} \otimes U^{T}) .

The inverse of $U^{*}$ is $U^{T}$ , not $U^{†}$ in general.

Displayed formulas in this passage

unlabeled · line 1230
$T : E D ⟶ D C^{'} D^{'} M, T = (I_{D} \otimes U^{*}) (∣ Φ_{m} ⟩_{F M} \otimes I_{E D}) .$
unlabeled · line 1237
$P = T T^{†}, Q = ∣ Φ_{δ} ⟩ ⟨ Φ_{δ} ∣_{D D^{'}} \otimes I_{C^{'} M} .$
unlabeled · line 1244
$T^{†} QT = L^{†} L = \frac{1}{m ^{2}} S .$
unlabeled · line 1249
$I - 2 P = (I_{D} \otimes U^{*}) [I - 2 (∣ Φ_{m} ⟩ ⟨ Φ_{m} ∣_{F M} \otimes I_{D E})] (I_{D} \otimes U^{T}) .$