Since the nineteenth century, the paradox of Maxwell's demon has puzzled numerous researchers concerning the foundation of the second law of thermodynamics . The demon seems to be able to extract a positive amount of work from an isothermal cycle. What reconciles the demon with the second law? Historically, a crucial observation was made by Landauer and Bennett; an excess energy cost is needed for information erasure of the demon's memory, because the erasure is logically irreversible. While the Landauer-Bennett resolution was widely accepted, several researchers have cast doubts on that argument . In this talk, I'd like to clarify how the paradox has been resolved based on the modern statistical mechanics and quantum information theory. Starting from the historical review of the paradox of Maxwell's demon, I will discuss the connection between quantum statistical mechanics and quantum information theory. The general properties of CPTP maps straightforwardly lead to the second law of thermodynamics, which states that the entropy production is non-negativce. The Landauer principle is shown to be a special case of the usual second law. In contrast, the entropy production can be negative for feedback-controlled processes by the demon, and must be definitely-positive for measurement processes . Therefore, an additional entropic cost is needed for the measurement process, which compensates for the negative entropy production with feedback control.
 "Maxwell's demon 2: Entropy, Classical and Quantum Information, Computing", H. S. Leff and A. F. Rex (eds.), (Princeton University Press, New Jersey, 2003).
 O. J. E. Maroney, "Information Processing and Thermodynamic Entropy", The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.).
 T. Sagawa and M. Ueda, Phys. Rev. Lett. 102, 250602 (2009); 106, 189901(E) (2011).