Distributed reconstruction with a specified link service

This section is an extension of, not a premise for, the local results.

Alice holds positions $1,2$; Bob holds position $3$, where the logical output is required. They may perform arbitrary local operations. There is no auxiliary shared entanglement, no quantum transfer, and no Bob-to-Alice message. Alice consumes her two input qubits and sends one classical message to Bob. Timing is not a free communication channel.

The streaming programme already gives a distributed decoder: Alice applies $B_{12}$, reads $(t,s)$, and sends those two bits. Bob applies $R_3,C_{ts}$ and returns position 3.