Thermodynamic Constraints on Quantum Information Gain and Error-Correction.

Quantum error correction (QEC) is a procedure by which the quantum state of a system is protected against a known type of noise, by preemptively adding redundancy to that state. Such a procedure is commonly applied in quantum computing when thermal noise is present. Interestingly, thermal noise has also been known to play a central role in quantum thermodynamics (QTD). This fact hints at the applicability of certain QTD statements in the QEC of thermal noise, which has been discussed previously in the context of Maxwell's demon. In this article, we view QEC as a quantum heat engine with a feedback controller (demon). The main task of this engine is to correct the effects of the hot bath (thermal noise) by attempting to close its own cycle with respect to the system state, corresponding to a perfect QEC. We derive an upper bound to the measurement heat that is dissipated during the error identification stage, thereby granting a physical meaning to negative values of quantum information gain. We also derive the second law of thermodynamics in the context of this QEC engine, operating with general quantum measurements. Finally, we show that, under a set of physically motivated assumptions, this leads to a fundamental trade-off relation between the fidelity of the QEC and the super-Carnot efficiencies that heat engines with feedback controllers have been known to possess.