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

Hamiltonian, code, and energy sectors

Hamiltonian, code, and energy sectors · lines 352–427

Hawk02-B@2#eq:S, Hawk02-B@2#thm:B1, Hawk02-B@2#eq:Uj, Hawk02-B@2#eq:rotation

Fix $d \geq 2$ , $N \geq 1$ , and $ω > 0$ . Let

H_{M} = span {∣ 0 ⟩, ∣ 1 ⟩, \dots, ∣ d ⟩}, H_{R_{j}} ≅ H_{M} .

The code $H_{C} = C^{d}$ has embedding

J_{C} ∣ a ⟩_{C} = ∣ a ⟩_{M}, 1 \leq a \leq d .

Set

H_{M} = ω a = 1 \sum^{d} ∣ a ⟩ ⟨ a ∣_{M}, H_{R_{j}} = ω a = 1 \sum^{d} ∣ a ⟩ ⟨ a ∣_{R_{j}}, H_{0} = H_{M} + j \sum H_{R_{j}} .

Each radiation cell begins in $∣ 0 ⟩$ . Every code state therefore has bare energy $ω$ .

Let $I_{j}$ be chronologically ordered disjoint intervals. The real integrable pulse $g_{j}$ is supported on $I_{j}$ and prescribed independently of the code and reference. In the interaction picture, define

H_{int, j} (t) = g_{j} (t) S_{j},

where

S_{j} = a = 1 \sum^{d} (∣ 0 ⟩ ⟨ a ∣_{M} \otimes ∣ a ⟩ ⟨ 0 ∣_{R_{j}} + ∣ a ⟩ ⟨ 0 ∣_{M} \otimes ∣ 0 ⟩ ⟨ a ∣_{R_{j}}) .

Other cells are acted on by the identity. Let

θ_{j} = \int_{I_{j}} g_{j} (t) d t, 0 \leq θ_{j} \leq π /2.

Previously emitted radiation does not reinteract.

The interval propagator is

U_{j} = e^{- i θ_{j} S_{j}} = I + (cos θ_{j} - 1) P_{j} - i sin θ_{j} S_{j},

where

P_{j} = S_{j}^{2} = a = 1 \sum^{d} (∣ a, 0 ⟩ ⟨ a, 0 ∣ + ∣ 0, a ⟩ ⟨ 0, a ∣)_{M R_{j}} .

It is unitary on the full space and satisfies

[S_{j}, H_{0}] = [U_{j}, H_{0}] = 0.

In particular,

U_{j} ∣ a ⟩_{M} ∣ 0 ⟩_{R_{j}} U_{j} ∣ 0 ⟩_{M} ∣ 0 ⟩_{R_{j}} = cos θ_{j} ∣ a ⟩_{M} ∣ 0 ⟩_{R_{j}} - i sin θ_{j} ∣ 0 ⟩_{M} ∣ a ⟩_{R_{j}}, = ∣ 0 ⟩_{M} ∣ 0 ⟩_{R_{j}} .

The sequence acts linearly on arbitrary reference-entangled inputs.

Displayed formulas in this passage

Hawk02-B@2#eq:S · line 382
$S_{j} = a = 1 \sum^{d} (∣ 0 ⟩ ⟨ a ∣_{M} \otimes ∣ a ⟩ ⟨ 0 ∣_{R_{j}} + ∣ a ⟩ ⟨ 0 ∣_{M} \otimes ∣ 0 ⟩ ⟨ a ∣_{R_{j}}) .$
Hawk02-B@2#eq:Uj · line 401
$U_{j} = e^{- i θ_{j} S_{j}} = I + (cos θ_{j} - 1) P_{j} - i sin θ_{j} S_{j},$
Hawk02-B@2#eq:rotation · line 417
$U_{j} ∣ a ⟩_{M} ∣ 0 ⟩_{R_{j}} U_{j} ∣ 0 ⟩_{M} ∣ 0 ⟩_{R_{j}} = cos θ_{j} ∣ a ⟩_{M} ∣ 0 ⟩_{R_{j}} - i sin θ_{j} ∣ 0 ⟩_{M} ∣ a ⟩_{R_{j}}, = ∣ 0 ⟩_{M} ∣ 0 ⟩_{R_{j}} .$
unlabeled · line 355
$H_{M} = span {∣ 0 ⟩, ∣ 1 ⟩, \dots, ∣ d ⟩}, H_{R_{j}} ≅ H_{M} .$
unlabeled · line 361
$J_{C} ∣ a ⟩_{C} = ∣ a ⟩_{M}, 1 \leq a \leq d .$
unlabeled · line 365
$H_{M} = ω a = 1 \sum^{d} ∣ a ⟩ ⟨ a ∣_{M}, H_{R_{j}} = ω a = 1 \sum^{d} ∣ a ⟩ ⟨ a ∣_{R_{j}}, H_{0} = H_{M} + j \sum H_{R_{j}} .$
unlabeled · line 378
$H_{int, j} (t) = g_{j} (t) S_{j},$
unlabeled · line 392
$θ_{j} = \int_{I_{j}} g_{j} (t) d t, 0 \leq θ_{j} \leq π /2.$
unlabeled · line 407
$P_{j} = S_{j}^{2} = a = 1 \sum^{d} (∣ a, 0 ⟩ ⟨ a, 0 ∣ + ∣ 0, a ⟩ ⟨ 0, a ∣)_{M R_{j}} .$
unlabeled · line 413
$[S_{j}, H_{0}] = [U_{j}, H_{0}] = 0.$

Hamiltonian, code, and energy sectors

Displayed formulas in this passage

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