Three independent ledgers

The three ledgers have different defining data:

1. Complete energy. A declared total Hamiltonian and all charged systems and controls are needed. For example, $U^{†}{H}_{\mathrm{tot}}U=H_{\mathrm{tot}}$ implies

   $$\mathrm{Tr}(H_{\mathrm{tot}}U\rho U^{†})=\mathrm{Tr}(H_{\mathrm{tot}}\rho ).$$

   No such $H_{\mathrm{tot}}$ is supplied merely by a channel name.

2. Erasure or reset cost. Discard returns no register. A reset instead has output

   $${\mathrm{Reset}}_M(X)=\mathrm{Tr}(X)|0⟩⟨0{|}_M.$$

   A thermodynamic claim additionally needs a memory implementation, reservoir, initial correlations, controls, and a work or heat convention. Landauer's account and finite-reservoir refinements concern such specified physical packages, not just the equality of two reduced channels Landauer,ReebWolf.

3. Residual leakage. Equation (eq:leakage) measures the input dependence of the complete complementary channel. It is a channel property and need not equal an energy, a bit count, or an erasure cost.

For completeness, a simple witness makes the last ledger operational.