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The thermodynamic meaning of negative entropy

An Addendum to this article was published on 24 August 2011


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 Here we show that the standard formulation and implications of Landauer’s principle are no longer valid in the presence of quantum information.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.

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Figure 1: Erasure in quantum computation.
Figure 2: Erasure setting.
Figure 3: Erasure of a pure state.
Figure 4: Erasure of a fully mixed state and work extraction.
Figure 5: General erasure procedure.


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We thank R. Colbeck for discussions. We acknowledge support from the Swiss National Science Foundation (L.d.R., J.A., R.R. and O.D.; grant no. 200021-119868 and the NCCR QSIT), the Portuguese Fundação para a Ciência e Tecnologia (L.d.R.; grant no. SFRH/BD/43263/2008), the European Research Council (R.R.; grant no. 258932) and Singapore’s National Research Foundation and Ministry of Education (V.V.).

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The main ideas were developed by all authors. L.d.R., J.A. and R.R. formulated and proved the main technical claims. L.d.R. and J.A. wrote the manuscript.

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Correspondence to Lídia del Rio.

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The authors declare no competing financial interests.

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Rio, L., Åberg, J., Renner, R. et al. The thermodynamic meaning of negative entropy. Nature 474, 61–63 (2011).

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