Theorem: Entanglement-assisted sector transfer

Suppose the receiver at $Q$ holds one half of a Bell pair whose preparation and distribution were completed before the unknown input is supplied. Suppose also that:

1. a Bell-basis measurement at $P$ is available;

2. its two outcome bits can be delivered from $P$ to $Q$ in $m$ ticks;

3. the receiver can apply the corresponding Pauli correction.

If measurement and correction are each charged one separate tick, the unknown qubit is transported in $m+2$ ticks, including when it is entangled with an untouched reference. The transport is outside the original-metric cone whenever $\ell >r(m+2)$.

Proof. The Bell measurement has four equally weighted corrected branches. In each branch, the $Q$-half is $X^b{Z}^a|\chi ⟩$, up to the fixed Bell-basis phase convention. Delivery of $(a,b)$, followed by $Z^a{X}^b$, leaves $|\chi ⟩$ at $Q$. The corrected branch maps from input to output are $I/2$, up to phase; summing their four CP maps gives the identity channel. Therefore the calculation preserves every reference.

The charged stages are one measurement tick, $m$ communication ticks, and one correction tick. Comparison with the cone requires the stated distance inequality.