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arXiv:1009.1630 · Nature

The thermodynamic meaning of negative entropy

Lídia del Rio, Johan Åberg, Renato Renner, Oscar Dahlsten, Vlatko Vedral

Published Jun 1, 2011• 422 citations
arXivHTMLPDF
DOI
10.1038/nature10123

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The second law of classical equilibrium thermodynamics, based on the positivity of entropy production, asserts that any process occurs only in a direction that some information may be lost (flow out of the system) due to the irreversibility inside the system. However, any thermodynamic system can exhibit fluctuations in which negative entropy production may be observed. In particular, in stochastic quantum processes due to quantum correlations and also memory effects we may see the reversal energy flow (heat flow from the cold system to the hot system) and the backflow of information into the system that leads to the negativity of the entropy production which is an apparent violation of the Second Law. In order to resolve this apparent violation, we will try to properly extend the Second Law to quantum processes by incorporating information explicitly into the Second Law. We will also provide a thermodynamic operational meaning for the flow and backflow of information. Finally, it is shown that negative and positive entropy production can be described by a quantum thermodynamic force.
Abstract The erasure of information is fundamentally an irreversible logical operation, carrying profound consequences for the energetics of computation and information processing. We investigate the thermodynamic costs associated with erasing (and preparing) quantum processes. Specifically, we analyze an arbitrary bipartite unitary gate acting on logical and ancillary input-output systems, where the ancillary input is always initialized in the ground state. We focus on the adversarial erasure cost of the reduced dynamics—that is, the minimal thermodynamic work cost to erase the logical output of the gate for any logical input, assuming full access to the ancilla but no access to any purifying reference of the logical input state. We determine that this adversarial erasure cost is directly proportional to the negative min-entropy of the reduced dynamics, thereby giving the dynamical min-entropy a clear operational meaning. The dynamical min-entropy can take positive and negative values, depending on the underlying quantum dynamics. The negative value of the erasure cost implies that the extraction of thermodynamic work is possible instead of its consumption during the process. A key foundation of this result is the quantum process decoupling theorem, which quantitatively relates the decoupling ability of a process with its min-entropy. This insight bridges thermodynamics, information theory, and the fundamental limits of quantum computation.
Irreversibility is characterized by the entropy production. The second law of classical thermodynamics, based on the positivity of the entropy production, only holds for deterministic processes. Therefore the Second Law in quantum thermodynamics, which is a stochastic thermodynamics, may not hold. By making a fundamental connection between thermodynamics and information theory we will introduce a new way of defining the Second Law which holds for both classical and quantum thermodynamics. Our work incorporate information well into the Second Law and also provide a thermodynamic operational meaning for negative and positive entropy production.