  We study three aspects of work statistics in the context of the fluctuation
theorem for the quantum spin chains up to $1024$ sites by numerical methods
based on matrix-product states (MPS). First, we use our numerical method to
evaluate the moments/cumulants of work done by sudden quench process on the
Ising or Haldane spin chains and study their behaviors across the quantum phase
transitions. Our results show that, up to the fourth cumulant, the work
statistics can indicate the quantum phase transition characterized by the local
order parameters but barely for purely topological phase transitions. Second,
we propose to use the fluctuation theorem, such as Jarzynski's equality, which
relates the real-time correlator to the ratio of the thermal partition
functions, as a benchmark indicator for the numerical real-time evolving
methods. Third, we study the passivity of ground and thermal states of quantum
spin chains under some cyclic impulse processes. We show that the passivity of
thermal states and ground states under the hermitian actions are ensured by the
second laws and variational principles, respectively, and also verify it by
numerical calculations. Besides, we also consider the passivity of ground
states under non-hermitian actions, for which the variational principle cannot
be applied. Despite that, we find no violation of passivity from our numerical
results for all the cases considered in the Ising and Haldane chains. {Overall,
we demonstrate that the work statistics for the sudden quench and impulse
processes can be evaluated precisely by the numerical MPS method to
characterize quantum phase transitions and examine the passivity of quantum
states. We also propose to exploit the universality of the fluctuation theorem
to benchmark the numerical real-time evolutions in an algorithm and model
independent way.
