Manuscript Oai069-A@1 · Oai069-A.tex · section · a4913aab

The concrete specification used here

Phase 1: fix the information task › The concrete specification used here · lines 348–394

Oai069-A@1#eq:task-algebras, Oai069-A@1#eq:target

Fix

E = C / (Z + i Z), d = 2^{s}, s \geq 1, g = 2^{h}, h \geq 0,

and put $ω = e^{2 π i / d}$ .

The named object is the rank-one local system $L_{1}$ constructed in Section sec:abelian. The finite roster includes all $L_{k}$ , $k \in Z / d Z$ .

Its quantum multiplicity is

C = C^{2},

and its gauge factor is $K = C^{g}$ . The input and target algebras are

A_{in} = k = 0 ⨁^{d - 1} B (C \otimes K), A_{tar} = k = 0 ⨁^{d - 1} B (C) .

Their traces are the sums of the ordinary block traces.

The complete target is

T ((ρ_{k})_{k}) = k \sum ∣ k ⟩ ⟨ k ∣_{J} \otimes Tr_{K} ρ_{k} .

Here $J$ is the returned sector label, represented by an orthogonal register. Its output is classical in that basis.

Every reference input is of the form

ρ_{Q, in} = k ⨁ ρ_{QC K, k}, ρ_{QC K, k} \geq 0, k \sum Tr ρ_{QC K, k} = 1.

Inside each block, $Q, C, K$ may be arbitrarily correlated. There is no independent-gauge preparation hypothesis.

Displayed formulas in this passage

Oai069-A@1#eq:task-algebras · line 368
$A_{in} = k = 0 ⨁^{d - 1} B (C \otimes K), A_{tar} = k = 0 ⨁^{d - 1} B (C) .$
Oai069-A@1#eq:target · line 377
$T ((ρ_{k})_{k}) = k \sum ∣ k ⟩ ⟨ k ∣_{J} \otimes Tr_{K} ρ_{k} .$
unlabeled · line 351
$E = C / (Z + i Z), d = 2^{s}, s \geq 1, g = 2^{h}, h \geq 0,$
unlabeled · line 363
$C = C^{2},$
unlabeled · line 386
$ρ_{Q, in} = k ⨁ ρ_{QC K, k}, ρ_{QC K, k} \geq 0, k \sum Tr ρ_{QC K, k} = 1.$