  In this paper we introduce a definition for conditional energy changes due to
general quantum measurements, as the change in the conditional energy evaluated
before, and after, the measurement process. By imposing minimal physical
requirements on these conditional energies, we show that the most general
expression for the conditional energy after the measurement is simply the
expected value of the Hamiltonian given the post-measurement state. Conversely,
the conditional energy before the measurement process is shown to be given by
the real component of the weak value of the Hamiltonian. Our definition
generalises well-known notions of distributions of internal energy change, such
as that given by stochastic thermodynamics. By determining the conditional
energy change of both system and measurement apparatus, we obtain the full
conditional work statistics of quantum measurements, and show that this
vanishes for all measurement outcomes if the measurement process conserves the
total energy. Additionally, by incorporating the measurement process within a
cyclic heat engine, we quantify the non-recoverable work due to measurements.
This is shown to always be non-negative, thus satisfying the second law, and
will be independent of the apparatus specifics for two classes of projective
measurements.
