Proposition: Thermal entropy-balance identity

Let a memory $M$ and reservoir $L$ initially have the product state ${\rho }_M\otimes {\gamma }_L$, where

After a specified unitary on $ML$, write the state as ${{\rho }^{\prime }}_{ML}$ and define

Then

Here $D$ is quantum relative entropy. This proposition neither constructs a reset operation nor optimizes its work or cyclic resource cost.

Proof. Unitary entropy preservation and the product preparation give

Hence $S({{\rho }^{\prime }}_L)-S({\gamma }_L)=\Delta S_M+I(M:L{)}_{{\rho }^{\prime }}$. Using $\log {\gamma }_L=-\beta H_L-\log Z$ gives

Combining the identities proves the equality. Mutual information and relative entropy are nonnegative. The operation and its resource constraints are hypotheses, not outputs of the calculation.