  The increase in average energy of a quantum system undergoing projective
energy measurements is referred to as "quantum heat", which is always zero. In
the framework of quantum stochastic thermodynamics, this is constructed as the
average over the fluctuating quantum heat (FQH), defined as the increase in
expected value of the Hamiltonian along two-point eigenstate trajectories.
However, such a definition has two drawbacks: (i) if the initial state does not
commute with the Hamiltonian and has degeneracies, the higher moments of the
FQH will not be uniquely defined, and therefore it is arguable whether such a
quantity is physically meaningful; (ii) the definition is operationally
demanding as it requires full knowledge of the initial state. In the present
manuscript we show that the FQH is an instance of conditional increase in
energy given sequential measurements, the first of which is with respect to the
eigen-decomposition of the initial state. By coarse-graining this initial
measurement, first by only distinguishing between degenerate subspaces of the
state, and finally by not distinguishing between any subspace at all, we
provide two alternative definitions for the FQH, which we call the partially
coarse-grained FQH and fully coarse-grained FQH, respectively. The partially
coarse-grained FQH resolves issue (i), whereas the fully coarse-grained FQH
resolves both (i) and (ii).
