Energy conservation and fluctuation theorem are incompatible for quantum work

Karen V. Hovhannisyan1 and Alberto Imparato2

1University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany
2Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus, Denmark

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Abstract

Characterizing fluctuations of work in coherent quantum systems is notoriously problematic. Here we reveal the ultimate source of the problem by proving that ($\mathfrak{A}$) energy conservation and ($\mathfrak{B}$) the Jarzynski fluctuation theorem cannot be observed at the same time. Condition $\mathfrak{A}$ stipulates that, for any initial state of the system, the measured average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange $deterministically$ zero work. Condition $\mathfrak{B}$ is only for thermal initial states and encapsulates the second law of thermodynamics. We prove that $\mathfrak{A}$ and $\mathfrak{B}$ are incompatible for work measurement schemes that are differentiable functions of the state and satisfy two mild structural constraints. This covers all existing schemes and leaves the theoretical possibility of jointly observing $\mathfrak{A}$ and $\mathfrak{B}$ open only for a narrow class of exotic schemes. For the special but important case of state-independent schemes, the situation is much more rigid: we prove that, essentially, only the two-point measurement scheme is compatible with $\mathfrak{B}$.

Work is one of the most basic notions in physics — it quantifies the transactions of “ordered” mechanical energy. At macroscopic scale, work is justifiably perceived as deterministic. However, at the microscale, especially in the quantum regime, significant fluctuations do occur, and they matter. Therefore, no characterization of energy consumption or storage capability of a quantum device will be complete without specifying the full statistics of consumed or delivered work.

Classically, the statistics of work are defined unambiguously: a deterministic value of work corresponds to each phase-space trajectory, and those trajectories have a well-defined probability distribution. In quantum mechanics, there are famously no phase-space trajectories, and therefore a qualitatively new philosophy is required for accessing the statistics of work. Many attempts have been made in the literature in that direction, but all of them eventually run into problems. Most commonly, issues arise whenever quantum coherence is to be accounted for. It has remained unclear whether a satisfactory approach to measuring work would ultimately be found.

We give a compelling resolution to this long-standing problem by showing that the two most fundamental laws associated with work — energy conservation and fluctuation theorem — cannot be observed simultaneously. Namely, no quantum measurement can yield results that would satisfy both laws. This means that any measurement of work inevitably comes with a trade-off, and therefore, each time a statement is made about work in the quantum regime, it must come with a disclaimer stipulating for which measurement scheme it is expected to be true.

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