Registers, names, and interfaces

Fix disjoint finite sets of quantum names $C$ and $S$, with

Their Hilbert spaces are

A register $Q$ denotes an arbitrary finite untouched reference. It may be entangled with $C\otimes S$. It is not available to the clock-reading program.

Write

The output record algebra is the diagonal algebra in the distinguished basis $\{|k{⟩}_R{\}}_{k\in \mathsf{Z}}$. Thus a recorded output has the form

This is a classical–quantum state, not a coherent quantum return of the clock.

The frozen operational interfaces are:

1. a preparation of ${\rho }_{CS}$, or a declared preparation channel with this output;

2. the accessible clock input $C$;

3. the surviving system output $S$;

4. the retained classical record $R$;

5. all untouched references, acted on by ${\mathrm{id}}_Q$.

If a purification, auxiliary token, encoding apparatus, or fresh clock supply is used, its ownership and access are declared separately. Nothing in the readout theorem acquires a state-dependent expectation token.