  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 used 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 (i.e., a demon). We derive an upper bound on the measurement heat
dissipated during the error-identification stage in terms of the Groenewold
information gain, thereby providing the latter with a physical meaning also
when it is negative. Further, we 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 triple trade-off relation, which implies a trade-off
between the maximum achievable fidelity of QEC and the super-Carnot efficiency
that heat engines with feedback controllers have been known to possess. A
similar trade-off relation occurs for the thermodynamic efficiency of the QEC
engine and the efficacy of the quantum measurement used for error
identification.
