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The quantum null energy condition (QNEC) is a lower bound on theenergy-momentum tensor in terms of the variation of the entanglement entropy ofa sub-region along a null direction. To gain insights into quantumthermodynamics of many-body systems, we study if the QNEC restrictsirreversible entropy production in quenches driven by energy-momentum inflowfrom an infinite memoryless bath in two-dimensional holographic theories. Wefind that an increase in both entropy and temperature, as implied by theClausius inequality of classical thermodynamics, are necessary but notsufficient to not violate QNEC in quenches leading to transitions betweenthermal states with momentum which are dual to Banados-Teitelboim-Zanelligeometries. For an arbitrary initial state, we can determine the lower andupper bounds on the increase of entropy (temperature) for a fixed increase intemperature (entropy). Our results provide explicit instances of quantum lowerand upper bounds on irreversible entropy production whose existence has beenestablished in literature. We also find monotonic behavior of thenon-saturation of the QNEC with time after a quench, and analytically determinetheir asymptotic values. Our study shows that the entanglement entropy of aninterval of length $l$ always thermalizes in time $l/2$ with an exponent $3/2$.Furthermore, we determine the coefficient of initial quadratic growth ofentanglement analytically for any $l$, and show that the slope of theasymptotic ballistic growth of entanglement for a semi-infinite interval istwice the difference of the entropy densities of the final and initial states.We determine explicit upper and lower bounds on these rates of growth ofentanglement.

Quantum thermodynamics of holographic quenches and bounds on the growth of entanglement from the QNEC