  The Kirkwood-Dirac quasiprobability distribution emerges from the quantum
correlation function of two observables measured at distinct times and is
therefore relevant for fundamental physics and quantum technologies. These
quasiprobabilities follow all but one of Kolmogorov axioms for joint
probability distributions: they can take non-positive values. Their
experimental reconstruction becomes challenging when expectation values of
incompatible observables are involved. Previous strategies aimed to reconstruct
them using weak measurements or combining strong measurements. Here, we use a
more direct approach, an interferometric scheme aided by an auxiliary system,
to reconstruct the Kirkwood-Dirac quasiprobability distribution. We
experimentally demonstrate the interferometric scheme in an electron-nuclear
spin system associated with a nitrogen-vacancy center in diamond. By measuring
the characteristic function, we reconstruct the quasiprobability distribution
of the work and analyze the behavior of the first and second moments of work.
Our results clarify the physical meaning of the work quasiprobability
distribution in the context of quantum thermodynamics. Finally, having measured
the real and imaginary parts of the Kirkwood-Dirac quasiprobability of work, we
are also able to study the uncertainty of measuring the Hamiltonian of the
system at two times, via the Robertson-Schr{\"o}dinger uncertainty relation,
for different initial states.
