  In this review paper, we discuss the statistical description in
non-equilibrium regimes of energy fluctuations originated by the interaction
between a quantum system and a measurement apparatus applying a sequence of
repeated quantum measurements. To properly quantify the information about
energy fluctuations, both the exchanged heat probability density function and
the corresponding characteristic function are derived and interpreted. Then, we
discuss the conditions allowing for the validity of the fluctuation theorem in
Jarzynski form $\langle e^{-\beta Q}\rangle = 1$, thus showing that the
fluctuation relation is robust against the presence of randomness in the time
intervals between measurements. Moreover, also the late-time, asymptotic
properties of the heat characteristic function are analyzed, in the
thermodynamic limit of many intermediate quantum measurements. In such a limit,
the quantum system tends to the maximally mixed state (thus corresponding to a
thermal state with infinite temperature) unless the system's Hamiltonian and
the intermediate measurement observable share a common invariant subspace.
Then, in this context, we also discuss how energy fluctuation relations change
when the system operates in the quantum Zeno regime. Finally, the theoretical
results are illustrated for the special cases of two- and three-levels quantum
systems, now ubiquitous for quantum applications and technologies.
