The thermodynamic meaning of negative entropy

Landauer's erasure principle, a widely accepted part of classical information theory first proposed by Rolf Landauer in 1961, asserts that it is necessary to perform work in order to erase data. This occurs when carrying out irreversible operations, thus releasing heat to the environment. For example, in electronics, heat generation is a major obstacle to circuitry miniaturization. Del Rio et al. show that the situation is completely different in the presence of quantum information about the system, and the implications of Landauer's principle are invalid. The more that is known about a system, the less it costs to erase it. An observer who is strongly correlated with a system may even gain work while erasing it, therefore cooling the environment. The quantum systems needed to experimentally demonstrate these results are, in principle, accessible with current technology. The heat generated by computations is not only an obstacle to circuit miniaturization but also a fundamental aspect of the relationship between information theory and thermodynamics. In principle, reversible operations may be performed at no energy cost; given that irreversible computations can always be decomposed into reversible operations followed by the erasure of data1,2, the problem of calculating their energy cost is reduced to the study of erasure. Landauer's principle states that the erasure of data stored in a system has an inherent work cost and therefore dissipates heat3,4,5,6,7,8. However, this consideration assumes that the information about the system to be erased is classical, and does not extend to the general case where an observer may have quantum information about the system to be erased, for instance by means of a quantum memory entangled with the system. Here we show that the standard formulation and implications of Landauer's principle are no longer valid in the presence of quantum information. Our main result is that the work cost of erasure is determined by the entropy of the system, conditioned on the quantum information an observer has about it. In other words, the more an observer knows about the system, the less it costs to erase it. This result gives a direct thermodynamic significance to conditional entropies, originally introduced in information theory. Furthermore, it provides new bounds on the heat generation of computations: because conditional entropies can become negative in the quantum case, an observer who is strongly correlated with a system may gain work while erasing it, thereby cooling the environment.