  Fluctuations of thermodynamic observables, such as heat and work, contain
relevant information on the underlying physical process. These fluctuations are
however not taken into account in the traditional laws of thermodynamics. While
the second law is extended to fluctuating systems by the celebrated fluctuation
theorems, the first law is generally believed to hold even in the presence of
fluctuations. Here we show that in the presence of quantum fluctuations, also
the first law of thermodynamics may break down. This happens because quantum
mechanics imposes constraints on the knowledge of heat and work. To illustrate
our results, we provide a detailed case-study of work and heat fluctuations in
a quantum heat engine based on a circuit QED architecture. We find
probabilistic violations of the first law and show that they are closely
connected to quantum signatures related to negative quasi-probabilities. Our
results imply that in the presence of quantum fluctuations, the first law of
thermodynamics may not be applicable to individual experimental runs.
