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

Complete emission isometry and both channels

Complete emission isometry and both channels · lines 457–519

Hawk02-B@2#eq:amplitudes, Hawk02-B@2#eq:L, Hawk02-B@2#thm:B2, Hawk02-B@2#eq:Wunnormalized, Hawk02-B@2#eq:W, Hawk02-B@2#eq:Nall, Hawk02-B@2#eq:C, Hawk02-B@2#eq:N

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Write

∣ vac ⟩_{R} = ∣ 0 ⟩_{R_{1}} \dots ∣ 0 ⟩_{R_{N}},

and let $∣ a; j ⟩$ denote excitation label $a$ in cell $j$ and vacuum elsewhere. Define

s_{N} = j \prod cos θ_{j}, q_{N} = s_{N}^{2}, α_{j} = - i sin θ_{j} k < j \prod cos θ_{k} .

Introduce the unnormalized emission operator

L_{N} ∣ a ⟩_{C} = j \sum α_{j} ∣ a; j ⟩_{R} .

The Hamiltonian sequence induces

W_{N} ∣ a ⟩_{C} = s_{N} ∣ a ⟩_{M} ∣ vac ⟩_{R} + ∣ 0 ⟩_{M} L_{N} ∣ a ⟩_{C} .

Furthermore,

L_{N}^{†} L_{N} = (1 - q_{N}) I_{C} .

When $q_{N} < 1$ , the normalized embedding

J_{N} = (1 - q_{N})^{- 1/2} L_{N}

has orthonormal columns and

W_{N} = q_{N} J_{C} \otimes ∣ vac ⟩ + 1 - q_{N} ∣ 0 ⟩_{M} \otimes J_{N} .

With complete ideal outputs $A = R_{1} \dots R_{N}$ and $B = M$ , the two channels, for all $q_{N}$ , are

N_{N} (X) C_{N} (X) = L_{N} X L_{N}^{†} + q_{N} Tr (X) ∣ vac ⟩ ⟨ vac ∣, = q_{N} J_{C} X J_{C}^{†} + (1 - q_{N}) Tr (X) ∣ 0 ⟩ ⟨ 0 ∣_{M} .

For $q_{N} < 1$ the accessible formula is equivalently

N_{N} (X) = (1 - q_{N}) J_{N} X J_{N}^{†} + q_{N} Tr (X) ∣ vac ⟩ ⟨ vac ∣ .

All formulas are identities on operators and on reference extensions.

Displayed formulas in this passage

Hawk02-B@2#eq:amplitudes · line 465
$s_{N} = j \prod cos θ_{j}, q_{N} = s_{N}^{2}, α_{j} = - i sin θ_{j} k < j \prod cos θ_{k} .$
Hawk02-B@2#eq:L · line 472
$L_{N} ∣ a ⟩_{C} = j \sum α_{j} ∣ a; j ⟩_{R} .$
Hawk02-B@2#eq:Wunnormalized · line 480
$W_{N} ∣ a ⟩_{C} = s_{N} ∣ a ⟩_{M} ∣ vac ⟩_{R} + ∣ 0 ⟩_{M} L_{N} ∣ a ⟩_{C} .$
Hawk02-B@2#eq:W · line 494
$W_{N} = q_{N} J_{C} \otimes ∣ vac ⟩ + 1 - q_{N} ∣ 0 ⟩_{M} \otimes J_{N} .$
Hawk02-B@2#eq:Nall · line 502
$N_{N} (X) C_{N} (X) = L_{N} X L_{N}^{†} + q_{N} Tr (X) ∣ vac ⟩ ⟨ vac ∣, = q_{N} J_{C} X J_{C}^{†} + (1 - q_{N}) Tr (X) ∣ 0 ⟩ ⟨ 0 ∣_{M} .$
Hawk02-B@2#eq:C · line 502
$N_{N} (X) C_{N} (X) = L_{N} X L_{N}^{†} + q_{N} Tr (X) ∣ vac ⟩ ⟨ vac ∣, = q_{N} J_{C} X J_{C}^{†} + (1 - q_{N}) Tr (X) ∣ 0 ⟩ ⟨ 0 ∣_{M} .$
Hawk02-B@2#eq:N · line 511
$N_{N} (X) = (1 - q_{N}) J_{N} X J_{N}^{†} + q_{N} Tr (X) ∣ vac ⟩ ⟨ vac ∣ .$
unlabeled · line 460
$∣ vac ⟩_{R} = ∣ 0 ⟩_{R_{1}} \dots ∣ 0 ⟩_{R_{N}},$
unlabeled · line 486
$L_{N}^{†} L_{N} = (1 - q_{N}) I_{C} .$
unlabeled · line 490
$J_{N} = (1 - q_{N})^{- 1/2} L_{N}$

Complete emission isometry and both channels

Displayed formulas in this passage

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