  The quantum null energy condition (QNEC) is a lower bound on the
energy-momentum tensor in terms of the variation of the entanglement entropy of
a sub-region along a null direction. To gain insights into quantum
thermodynamics of many-body systems, we study if the QNEC restricts
irreversible entropy production in quenches driven by energy-momentum inflow
from an infinite memoryless bath in two-dimensional holographic theories. We
find that an increase in both entropy and temperature, as implied by the
Clausius inequality of classical thermodynamics, are necessary but not
sufficient to not violate QNEC in quenches leading to transitions between
thermal states with momentum which are dual to Banados-Teitelboim-Zanelli
geometries. For an arbitrary initial state, we can determine the lower and
upper bounds on the increase of entropy (temperature) for a fixed increase in
temperature (entropy). Our results provide explicit instances of quantum lower
and upper bounds on irreversible entropy production whose existence has been
established in literature. We also find monotonic behavior of the
non-saturation of the QNEC with time after a quench, and analytically determine
their asymptotic values. Our study shows that the entanglement entropy of an
interval of length $l$ always thermalizes in time $l/2$ with an exponent $3/2$.
Furthermore, we determine the coefficient of initial quadratic growth of
entanglement analytically for any $l$, and show that the slope of the
asymptotic ballistic growth of entanglement for a semi-infinite interval is
twice the difference of the entropy densities of the final and initial states.
We determine explicit upper and lower bounds on these rates of growth of
entanglement.
