Proposition: Clock and record limits

The clock channel has no universal left inverse. Its classical record marginal does not determine the quantum residual. No finite clock matrices satisfy $[T_C,H_C]=i{I}_C$. Discarding the clock record gives ${\mathrm{Tr}}_{C}Z$.

Proof. With the same payload, the clock states

have identical destructive outputs. A left inverse cannot recover both. The inputs $P_0\otimes {\rho }_X$ and $P_0\otimes {\rho }_X^{\prime }$ have the same deterministic record and different residuals. The commutator claim follows by taking matrix traces. Finally, summing diagonal clock blocks gives the partial trace.