Proposition: Unitarity and recovery do not fix energy accounting

Equation (eq:unitary-energy) preserves the expectation of $H_{\mathrm{tot}}$ for every input state. Equation (eq:isometric-energy) preserves the corresponding input energy expectation. Neither property follows merely from $U^{†}U=I$ or $V^{†}V=I$. Neither property by itself implies outgoing recovery.

Proof. The first two statements follow by trace duality. For the nonimplication, a qubit bit flip is unitary but does not commute with $H=ϵ|1⟩⟨1|$ when $ϵ>0$. Failure of recovery despite exact conserving evolution is proved in Proposition prop:insufficiency.