Composition with readout and the table probe

Let ${\mathcal{T}}_m$ be the route channel with blanks included, and let ${\iota }_A:XQ\to A_m$ be the fixed output identification. For arbitrary ${\rho }_{CXE}$, the complete output of readout, local controlled probe preparation, and routing is

This follows by applying Theorem thm:transport to each subnormalized block and summing.

Reading the transported $Q$ in the $Z$ basis gives (eq:binary), with the residual $X$ relocated to $x_B$. Consequently the centered operational fit (eq:operational-fit) survives transport exactly.

The clock record $R_A$ remains at the sender in (eq:composed). A joint-state formula is not instantaneous delivery of that record. A receiver operation depending on $k$ requires a separately declared causal record wire, or the record-controlled operation must be performed before departure. Our controlled probe preparation is performed at the sender. The terminal $Z$ readout itself is independent of $k$. Joint records can be collected later in a common causal future.