Local effective dynamics of quantum systems: A generalized approach to work and heat

Abstract

By computing the local energy expectation values with respect to some local measurement basis we show that for any quantum system there are two fundamentally different contributions: changes in energy that do not alter the local von Neumann entropy and changes that do. We identify the former as work and the latter as heat. Since our derivation makes no assumptions on the system Hamiltonian or its state, the result is valid even for states arbitrarily far from equilibrium. Examples are discussed ranging from the classical limit to purely quantum mechanical scenarios, i.e. where the Hamiltonian and the density operator do not commute. The formulation of classical thermodynamics was one of the most important achievements of the 19 th century, as it allowed to investigate a large variety of phenomena, including the workings of thermodynamical machines. The first law of thermodynamics, dU =dW +dQ,combined with definitions for the infinitesimal change in workdW and heatdQ and the second law is all that is required for computing important quantities like the efficiency of a process. In the quantum realm, the classification of work and heat is less clear. So far, it has mainly been based on the change of the total energy expectation value dU = dTr Ĥρ = Tr ρdĤ +Ĥdρ ,and defining the first term asdW and the second asdQ [1][2][3][4][5]. However, such a classification can be problematic, for various reasons: For one, it is not obvious how to apply this definition to processes involving an internal transfer of work and heat, as is the case, e.g., in algorithmic cooling, a method to obtain highly polarized spins by applying a series of quantum gates [6]. Then, the microscopic foundation of (2) is rather unclear: As thermodynamic behavior may occur even in small quantum systems [7], it should, in principle, be possible to obtain dW anddQ even there. In the following, we will present an alternative definition that does not suffer from the problems above. This letter is organized as follows. We first discuss the local effective dynamics of a bipartite quantum system. This dynamics of one part of the system is a reduced dynamics, which depend on the state of and the interaction with the rest of the system. Because we are interested in local properties of a part of the system, it is necessary to get a complete local description. In contrast to, e.g., a Markovian quantum master equation (see, e.g., [8]) the reduced dynamics cannot expected to be a set of closed differential equations. Based upon what an experimentalist might observe, we give a definition for the local energy. We then show that the change in this local energy can always be split into a part that correlates with a change in entropy and in a part which does not. Corresponding to classical thermodynamics, the former will be called "heat" and the latter "work". However, this definition for the local heat and work does not only depend on local properties, but on details of the whole system. We explicitly give formulas to calculate those local quantities once the time evo...

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