  Thermodynamics constrains changes to the energy of a system, both deliberate
and random, via its first and second laws. When the system is not in
equilibrium, fluctuation theorems such as the Jarzynski equality further
restrict the distributions of deliberate work done. Such fluctuation theorems
have been experimentally verified in small, non-equilibrium quantum systems
undergoing unitary or decohering dynamics. Yet, their validity in systems
governed by a non-Hermitian Hamiltonian has long been contentious, due to the
false premise of the Hamiltonian's dual and equivalent roles in dynamics and
energetics. Here we show that work fluctuations in a non-Hermitian qubit obey
the Jarzynski equality even if its Hamiltonian has complex or purely imaginary
eigenvalues. With post-selection on a dissipative superconducting circuit
undergoing a cyclic parameter sweep, we experimentally quantify the work
distribution using projective energy measurements and show that the fate of the
Jarzynski equality is determined by the parity-time symmetry of, and the
energetics that result from, the corresponding non-Hermitian, Floquet
Hamiltonian. By distinguishing the energetics from non-Hermitian dynamics, our
results provide the recipe for investigating the non-equilibrium quantum
thermodynamics of such open systems.
