Theorem: B5: exact streaming memory optimum

Under Definition def:stream, exact subsystem recovery from the authorized triple requires at least two live laboratory qubits, even with unrestricted classical storage and arbitrary local single-qubit operations. Hence

Proof. Suppose the peak were one. Before the second carrier can be delivered, the first carrier must cease to be a live laboratory qubit. There is no early quantum output, return to the supplier, or other quantum storage. Its only possible continuing influence is therefore a classical record.

The implemented channel consequently has the form

with positive effects $E_h$ on position 1 and subsequent completely positive maps on positions 2 and 3. Its Choi operator is a sum of positive products across input-reference position 1 versus all remaining systems. It is separable across that cut and has positive partial transpose.

Exact recovery on the code would force $\Lambda =\mathcal{D}$ by Theorem thm:B3. Equation (eq:choi) has a negative partial transpose, a contradiction. Theorem thm:streamupper attains peak two.