Proposition: Clock limits and observation boundaries

The following hold:

1. ${\mathcal{M}}_C$ has no universal left inverse.

2. Its record marginal alone does not certify the residual.

3. Discarding its clock record gives ${\mathrm{Tr}}_R{\mathcal{M}}_C(Z)={\mathrm{Tr}}_{C}Z$, including references.

4. No finite matrices obey $[T_C,H_C]=i{I}_C$.

5. A record-only channel followed by a preparation decoder cannot recover every input of a quantum code of dimension at least two.

Proof. With the same payload, the two clock states

have identical destructive outputs, proving (i). Inputs $P_0\otimes {\rho }_X$ and $P_0\otimes {\rho }_X^{\prime }$ have the same deterministic record and different residuals, proving (ii). Summing diagonal clock blocks proves (iii). Taking traces of the proposed commutator proves (iv).

For (v), a channel factoring through a classical register has measure-and-prepare form

On one half of a maximally entangled code-reference state, its output is separable. A separable state has overlap at most $1/d<1$ with that maximally entangled state: on a product state the overlap is $\mathrm{Tr}({\sigma }^{\mathsf{T}}\tau )/d\le 1/d$, and the inequality is preserved by convex combinations. It cannot be the identity channel.