Manuscript Hawk01-B@1 · Hawk01-B.tex · section · cfa976f1

Exact matrix-entry recipe

An independently checkable small example › Exact matrix-entry recipe · lines 2072–2115

Hawk01-B@1#eq:matrixentries, Hawk01-B@1#eq:successentries, Hawk01-B@1#eq:failureentries

The transition maps themselves are the archive needed to reproduce the example. For general $d$ ,

(U_{a, b})_{k i} = ζ^{bi} 1 {k = i + a mod d} .

At prefix $t$ , nonzero entries of $W_{t}$ are:

⟨ j, r ∣_{B_{t}} ⟨ r, j, k ∣_{R} W_{t} ∣ i ⟩_{A} ⟨ f, r, k ∣_{B_{t}} ⟨ r, vac ∣_{R} W_{t} ∣ i ⟩_{A} = \frac{w _{j}}{d} (U_{r})_{k i}, = \frac{s _{t}}{d} (U_{r})_{k i} .

Every omitted entry is zero. The radiation labels in these equations include the all-other-vacuum pattern. The endpoint relocates the residual basis index into the endpoint radiation factor exactly as in (eq:endpointsuccess)–(eq:endpointfailure).

An independent implementation can verify:

$W_{t}^{†} W_{t} = I_{d}$ ;
$(H_{B_{t}} + H_{R_{\leq t}}) W_{t} = M W_{t}$ ;
the corresponding charge-intertwining identities;
the two partial traces (eq:accessibleprefix) and (eq:complementprefix);
the decoder identity (eq:decodedprefix);
the entropy eigenvalues used in Theorem thm:information;
the endpoint constant complement.

All table values are consequences of these exact formulas, not unreported numerical experiments.

Displayed formulas in this passage

Hawk01-B@1#eq:matrixentries · line 2076
$(U_{a, b})_{k i} = ζ^{bi} 1 {k = i + a mod d} .$
Hawk01-B@1#eq:successentries · line 2083
$⟨ j, r ∣_{B_{t}} ⟨ r, j, k ∣_{R} W_{t} ∣ i ⟩_{A} ⟨ f, r, k ∣_{B_{t}} ⟨ r, vac ∣_{R} W_{t} ∣ i ⟩_{A} = \frac{w _{j}}{d} (U_{r})_{k i}, = \frac{s _{t}}{d} (U_{r})_{k i} .$
Hawk01-B@1#eq:failureentries · line 2083
$⟨ j, r ∣_{B_{t}} ⟨ r, j, k ∣_{R} W_{t} ∣ i ⟩_{A} ⟨ f, r, k ∣_{B_{t}} ⟨ r, vac ∣_{R} W_{t} ∣ i ⟩_{A} = \frac{w _{j}}{d} (U_{r})_{k i}, = \frac{s _{t}}{d} (U_{r})_{k i} .$