The Pasternak Identity: Exact, Additive Cost Accounting in Quantum Thermodynamics

We introduce the Pasternak Identity as an exact identity that causally decomposes the scaled dissipated energy in finite‑dimensional quantum thermodynamics into three non‑negative and operationally meaningful contributions: the fundamental information‑condensation cost, a correlation overhead, and a thermalization overhead. From first principles—global unitarity and the properties of the quantum relative entropy—we prove two exact bridges whose synthesis yields the Pasternak Identity (PI). Unlike inequalities that provide only lower bounds, PI gives a complete, additive accounting and delivers necessary‑and‑sufficient conditions for thermodynamic reversibility. It generalizes additively to multiple baths and admits a strong‑coupling formulation via a mean‑force framework. We provide full derivations, address initially correlated states, connect the identity to quantum channel theory, and illustrate the trade-off between correlation and thermalization overheads via a qubit-reset example. Central identity (see Eq. (PI)): (valid for information condensation, i.e., ).