Thermodynamic analysis of quantum error-correcting engines

Quantum error-correcting codes can be cast in a way which is strikingly similar to a quantum heat engine undergoing an Otto cycle. In this paper, we strengthen this connection further by carrying out a complete assessment of the thermodynamic properties of four-stroke operator-based error-correcting codes. This includes an expression for the entropy production in the cycle which, as we show, contains clear contributions stemming from the different sources of irreversibility. To illustrate our results, we study a classical three-qubit error-correcting code, well suited for incoherent states, and the nine-qubit Shor code capable of handling fully quantum states. We show that the work cost associated with the correction gate is directly associated with the heat introduced by the error. Moreover, the work cost associated with encoding and decoding quantum information is always positive, a fact which is related to the intrinsic irreversibility introduced by the noise. Finally, we find that correcting the coherent (and thus genuinely quantum) part of a quantum state introduces substantial modifications related to the Hadamard gates required to encode and decode coherences.