  The exchange of energy between a classical open system and its environment
can be analysed for a single run of an experiment using the phase space
trajectory of the system. By contrast, in the quantum regime such energy
exchange processes must be defined for an {\it ensemble} of runs of the same
experiment based on the reduced system density matrix. Single-shot approaches
have been proposed for quantum systems that are weakly coupled to a heat bath.
However, a single-shot analysis for a quantum system that is entangled or
strongly interacting with external degrees of freedom has not been attempted
because no system wave function exists for such a system within the standard
formulation of quantum theory. Using the notion of the {\it conditional} wave
function of a quantum system, we derive here an exact formula for the rate of
total energy change in an open quantum system, valid for arbitrary coupling
between the system and the environment. In particular, this allows us to
identify three distinct contributions to the total energy flow: an external
contribution coming from the explicit time dependence of the Hamiltonian, an
interaction contribution associated with the interaction part of the
Hamiltonian, and an entanglement contribution, directly related to the presence
of entanglement between the system and its environment. Given the close
connection between weak values and the conditional wave function, the approach
presented here provides a new avenue for experimental studies of energy
fluctuations in open quantum systems.
