Proposition: No-return, record separation, and finite CCR

The following hold.

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

2. Its classical record marginal does not determine its quantum residual.

3. No finite-dimensional matrices satisfy $[T_C,H_C]=i{I}_C$.

Proof. The two clock states

with the same $X$ input give the same output under ${\mathcal{M}}_C$. A left inverse cannot recover both.

Inputs $P_0\otimes {\rho }_X$ and $P_0\otimes {\rho }_X^{\prime }$ have the same deterministic classical record but different residuals when ${\rho }_X\ne {\rho }_X^{\prime }$.

Finally, $\mathrm{Tr}[T_C,H_C]=0$, whereas $\mathrm{Tr}(i{I}_C)=4i$.