Quantum Thermodynamics of Holographic Quenches and Bounds on the Growth of Entanglement from the Quantum Null Energy Condition

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 subregion 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, is necessary but not sufficient to not violate QNEC in quenches leading to transitions between thermal states with momentums that 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 nonsaturation 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.

Figure 1

Schematic representation of the cut and glue method—the left and right halves represent the separate Poincaré patches to which the prequench and postquench spacetimes map to. The gluing hypersurfaces ${\mathrm{\Sigma }}^{i,f}$ are the images of $t=0$ in the respective geometries. Points on ${\mathrm{\Sigma }}^{i,f}$ carrying the same physical coordinate labels $x$ and $r$ are identified. The geodesic ending at the boundary is cut into three arcs.

Figure 2

Left: The possible relative changes $[({\mu }_{\pm }^f-{\mu }_{\pm }^i)/{\mu }_{\pm }^i]$ for ${\mu }_{+}^i=1$, ${\mu }_{-}^i=0.75$ are shown in white. The black lines are given by the inequality [Eq. (13)] required by ${\mathcal{Q}}_{\pm }\ge 0$ at $t=0$ and the gray dotted region is disallowed by examining ${\mathcal{Q}}_{-}$ for $t>0$. The contours show that for a fixed change in temperature (red), the change in entropy is bounded from above and below (yellow and blue). Right: The same allowed region (in white) is shown in terms of the relative change in temperature and entropy density.