  We analyse the nature of the statistics of the work done on or by a quantum
many-body system brought out of equilibrium. We show that, for the sudden
quench and for an initial state which commutes with the initial Hamiltonian, it
is possible to retrieve the whole non-equilibrium thermodynamics via single
projective measurements of observables. We highlight in a physically clear way
the qualitative implications for the statistics of work coming from considering
processes described by operators that either commute or do not commute with the
unperturbed Hamiltonian of a given system. We consider a quantum many-body
system and derive an expression that allows us to give a physical
interpretation, for a thermal initial state, to all of the cumulants of the
work in the case of quenched operators commuting with the unperturbed
Hamiltonian. In the commuting case the observables that we need to measure have
an intuitive physical meaning. Conversely, in the non-commuting case we show
that, although it is possible to operate fully within the single-measurement
framework irrespectively of the size of the quench, some difficulties are faced
in providing a clear-cut physical interpretation to the cumulants. This
circumstance makes the study of the physics of the system non-trivial and
highlights the non-intuitive phenomenology of the emergence of thermodynamics
from the fully quantum microscopic description. We illustrate our ideas with
the example of the Ising model in a transverse field showing the interesting
behaviour of the high-order statistical moments of the work distribution for a
generic thermal state and linking them to the critical nature of the model
itself.
