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An important result in classical stochastic thermodynamics is the workfluctuation--dissipation relation (FDR), which states that the dissipated workdone along a slow process is proportional to the resulting work fluctuations.Here we show that slowly driven quantum systems violate this FDR wheneverquantum coherence is generated along the protocol, and derive a quantumgeneralisation of the work FDR. The additional quantum terms in the FDR arefound to lead to a non-Gaussian work distribution. Fundamentally, our resultshows that quantum fluctuations prohibit finding slow protocols that minimiseboth dissipation and fluctuations simultaneously, in contrast to classical slowprocesses. Instead, we develop a quantum geometric framework to find processeswith an optimal trade-off between the two quantities.

Work fluctuations in slow processes: quantum signatures and optimal control