The resource theory of informational nonequilibrium in thermodynamics

We review recent work on the foundations of thermodynamics in the light of quantum information theory. We adopt a resource-theoretic perspective, wherein thermodynamics is formulated as a theory of what agents can achieve under a particular restriction, namely, that the only state preparations and transformations that they can implement for free are those that are thermal at some fixed temperature. States that are out of thermal equilibrium are the resources. We consider the special case of this theory wherein all systems have trivial Hamiltonians (that is, all of their energy levels are degenerate). In this case, the only free operations are those that add noise to the system (or implement a reversible evolution) and the only nonequilibrium states are states of informational nonequilibrium, that is, states that deviate from the maximally mixed state. The degree of this deviation we call the state's nonuniformity; it is the resource of interest here, the fuel that is consumed, for instance, in an erasure operation. We consider the different types of state conversion: exact and approximate, single-shot and asymptotic, catalytic and noncatalytic. In each case, we present the necessary and sufficient conditions for the conversion to be possible for any pair of states, emphasizing a geometrical representation of the conditions in terms of Lorenz curves. We also review the problem of quantifying the nonuniformity of a state, in particular through the use of generalized entropies, and that of quantifying the gap between the nonuniformity one must expend to achieve a single-shot state preparation or state conversion and the nonuniformity one can extract in the reverse operation. Quantum state-conversion problems in this resource theory can be shown to be always reducible to their classical counterparts, so that there are no inherently quantum-mechanical features arising in such problems. This body of work also demonstrates that the standard formulation of the second law of thermodynamics is inadequate as a criterion for deciding whether or not a given state transition is possible. (C) 2015 Elsevier B.V. All rights reserved.