Manuscript Hawk05-B@1 · Hawk05-B.tex · proposition · f2e5d4f5

Proposition: Complete conserving implementation

One coherent emission history with a Haar hypothesis › A nontrivial conserving realization · lines 499–531 · proof included or linked

Hawk05-B@1#prop:implementation, Hawk05-B@1#eq:Vk

Back to manuscript

The extended $U$ and all emission SWAPs commute with $H_{0}$ . At cut $k$ , the effective quantum output is

R_{k} = (C^{s})^{\otimes k}, B_{k} = (C^{s})^{\otimes (N - k)},

with

r_{k} = s^{k}, b_{k} = s^{N - k}, r_{k} b_{k} = D .

Up to fixed vacant factors and tensor permutations, its complete isometry is

V_{k}^{U} = F_{k} U J : C \to R_{k} \otimes B_{k},

where $F_{k}$ is the canonical tensor regrouping. Earlier emitted cells are never acted on again. At $k = N$ , the residual is a fixed pure vacant state.

Proof. The all-occupied memory sector has one $H_{0}$ eigenvalue, $N ω$ ; thus arbitrary $U$ on that sector commutes with $H_{0}$ . SWAP preserves $P_{M_{j}} + P_{R_{j}}$ . After $k$ swaps, the first $k$ internal tensor factors are in radiation, while the remaining factors are in memory. All other factors are fixed vacancies. This is (eq:Vk). At the last cut no internal memory factor remains.

Displayed formulas in this passage

Hawk05-B@1#eq:Vk · line 514
$V_{k}^{U} = F_{k} U J : C \to R_{k} \otimes B_{k},$
unlabeled · line 503
$R_{k} = (C^{s})^{\otimes k}, B_{k} = (C^{s})^{\otimes (N - k)},$
unlabeled · line 509
$r_{k} = s^{k}, b_{k} = s^{N - k}, r_{k} b_{k} = D .$

Proposition: Complete conserving implementation

Displayed formulas in this passage

Nearby in this section