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The exchange of energy between a classical open system and its environmentcan be analysed for a single run of an experiment using the phase spacetrajectory of the system. By contrast, in the quantum regime such energyexchange processes must be defined for an {\it ensemble} of runs of the sameexperiment based on the reduced system density matrix. Single-shot approacheshave 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 orstrongly interacting with external degrees of freedom has not been attemptedbecause no system wave function exists for such a system within the standardformulation of quantum theory. Using the notion of the {\it conditional} wavefunction of a quantum system, we derive here an exact formula for the rate oftotal energy change in an open quantum system, valid for arbitrary couplingbetween the system and the environment. In particular, this allows us toidentify three distinct contributions to the total energy flow: an externalcontribution coming from the explicit time dependence of the Hamiltonian, aninteraction contribution associated with the interaction part of theHamiltonian, and an entanglement contribution, directly related to the presenceof entanglement between the system and its environment. Given the closeconnection between weak values and the conditional wave function, the approachpresented here provides a new avenue for experimental studies of energyfluctuations in open quantum systems.

Contributions to single-shot energy exchanges in open quantum systems