Theorem: Serial teleportation with complete records

Suppose a path with $L$ hops supplies an independent Bell pair per hop and per payload qubit, local Bell measurement, classical outcome communication, and the corresponding Pauli correction.

For an $n$-qubit payload, the deterministic complete channel after all corrections is

where $R$ is the retained $2nL$-bit record. This identity holds with every untouched reference.

The resource ledger includes $nL$ Bell pairs and $2nL$ transmitted classical bits. With fresh streamed Bell pairs, the active quantum interface can use $3n$ qubits per hop; a preprovisioned external Bell-pair pool must additionally be counted.

Proof. For one qubit and one hop, each corrected Bell-measurement branch has Kraus map $I/2$, up to an irrelevant branch phase. Its selective map is therefore $X/4$. Summing the four branches returns $X$, with a uniform independent classical record.

Tensoring over $n$ payload qubits and composing $L$ hops gives one selective map $X/4^{nL}$ for each record. The operator identity proves the reference statement. The consumption and resource counts follow from one payload block and the two halves of one resource pair at each active hop.