Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

Causal delivery of the classical code

Causal schedules, records, and payload transport › Causal delivery of the classical code · lines 2106–2136

The receiver's probe operation depends on $k$ . A joint state formula does not make $k$ available remotely. Install a two-bit classical record register at each route site, initially blank except at the sender. At tick $j$ , copy the classical code from site $j - 1$ to site $j$ , while retaining the earlier copy.

These are maps on diagonal classical algebras, not cloning maps for unknown quantum states. They have the same causally embedded endpoint dependencies as the payload route.

For the clock output

k \sum ∣ k ⟩ ⟨ k ∣_{R_{A}} \otimes σ_{X E, k},

the final state is

k \sum (j = 0 ⨂^{n} ∣ k ⟩ ⟨ k ∣_{R_{j}}) \otimes σ_{A_{n} E, k} \otimes j = 0 ⨂^{n - 1} ω_{A_{j}} .

The receiver now applies $M_{k, y}$ locally, using $R_{n}$ . The complete final instrument is (eq:complete-pilot), with its residual relocated to $x_{B}$ and with additional retained code copies and route blanks.

Thus Theorem thm:pilot survives transport exactly.

Displayed formulas in this passage

unlabeled · line 2120
$k \sum ∣ k ⟩ ⟨ k ∣_{R_{A}} \otimes σ_{X E, k},$
unlabeled · line 2124
$k \sum (j = 0 ⨂^{n} ∣ k ⟩ ⟨ k ∣_{R_{j}}) \otimes σ_{A_{n} E, k} \otimes j = 0 ⨂^{n - 1} ω_{A_{j}} .$