  Describing dynamics of quantum many-body systems is a formidable challenge
due to rapid generation of quantum entanglement between remote degrees of
freedom. A promising approach to tackle this challenge, which has been proposed
recently, is to characterize the quantum dynamics of a many-body system and its
properties as a bath via the Feynman-Vernon influence matrix (IM), which is an
operator in the space of time trajectories of local degrees of freedom.
Physical understanding of the general scaling of the IM's temporal entanglement
and its relation to basic dynamical properties is highly incomplete to present
day. In this Article, we analytically compute the exact IM for a family of
integrable Floquet models - the transverse-field kicked Ising chain - finding a
Bardeen-Cooper-Schrieffer-like "wavefunction" on the Schwinger-Keldysh contour
with algebraically decaying correlations. We demonstrate that the IM exhibits
area-law temporal entanglement scaling for all parameter values. Furthermore,
the entanglement pattern of the IM reveals the system's phase diagram,
exhibiting jumps across transitions between distinct Floquet phases. Near
criticality, a non-trivial scaling behavior of temporal entanglement is found.
The area-law temporal entanglement allows us to efficiently describe the
effects of sizeable integrability-breaking perturbations for long evolution
times by using matrix product state methods. This work shows that tensor
network methods are efficient in describing the effect of non-interacting baths
on open quantum systems, and provides a new approach to studying quantum
many-body systems with weakly broken integrability.
