Energy-accounting validation of QEC-assisted quantum batteries

Quantum error correction (QEC) protects logical information, whereas a quantum battery is judged by the work it can deliver. We ask when a recovery operation applied during storage may be credited with protecting a battery's ergotropy. The second law bounds the average work of each recovery call from below by the change of nonequilibrium free energy it causes at bath temperature T; as a storage convention we debit the nonnegative part of this change and take the net ergotropy gain left per call as the budget for all further overhead. We apply this ledger to batteries of two to four two-level cells under collective and independent Lindblad noise, testing a coherence-restoring heuristic, the four-qubit [[4,1,2]] amplitude-damping code with two recovery completions, standard channel families, and reset channels. Every state in the code space of the [[4,1,2]] code has zero mean energy and at most half the maximal ergotropy, so a decoder that restores the code space cannot protect a fully charged battery; a completion that leaves the rest of the Hilbert space untouched avoids this bound but still loses ergotropy on a fully charged battery during storage. Unital recoveries cannot raise the ergotropy by more than the energy they supply. For four cells and inputs diagonal in the energy basis, an optimal energy-nonincreasing covariant recovery is attained by a reset channel, whose value we give in closed form and whose gain comes from removing entropy. At 10 mK and 10 GHz the tested recoveries leave at most 3.3×10⁻² energy quanta per call beyond the debit, less than one photon at the cell frequency, so any implementation that dissipates a single quantum per call beyond the debit loses ergotropy.