Moving Mirrors, Page Curves, and Bulk Entropies in ${\mathrm{AdS}}_2$

Understanding the entanglement of radiation in quantum field theory has been a long-standing challenge, with implications ranging from black hole thermodynamics to quantum information. We demonstrate how the case of the free fermion in $1+1$ dimensions reveals the details of the density matrix of the radiation produced by a moving mirror. Using the resolvent method rather than standard conformal field theory techniques we derive the Rényi entropies, modular Hamiltonian, and flow of the radiation and determine when mirrors generate unitary transformations.

Figure 1

Evolution of an entangling region $V$ for a given mirror trajectory. The Rényi entropies of $V$ depend on the mirror only through the position and velocity at the null projections of the region’s boundary, as exemplified in (18). Thus if these return to their original values, the mirror produces unitary transformations.

Figure 2

Illustration of the modular evolution governed by (24), for two disjoint intervals and a static mirror and incoming vacuum. We plot (26) for equal and opposite chiralities (this is not a spacetime diagram). The modular flow of ${\psi }_i(x)$ yields, at each interval, contributions of both chiralities ${\psi }_{\pm i}$ located at the points $y_{\pm \ell }(x)$ that are solutions to (26) (colored circles). Evolution in modular time $\tau$ shifts the curves vertically, and the roots evolve accordingly. The root at $y_2$ is the local (geometric) solution, with $y_2\to x$ as $\tau \to 0$. Here $a_1=1$, $b_1=2$, $a_2=3$, $b_2=5.3$.

Figure 3

Two different mirror trajectories with entangling region $V=V_1\cup V_2$. Although the incoming and outgoing states ${\rho }_{\mathrm{in}/\mathrm{out}}$ are very different, unitarity means that their Rényi entropies on arbitrary $V$ match. Illustrated are $g_I$ and $g_{\mathrm{II}}$ from the main text: the former starts to accelerate, becoming asymptotically null, describing a nonunitary process. The latter follows the former for some time before returning to its original trajectory and respects unitarity.

Figure 4

Evolution of mutual information (20) between the regions $V_1$ and $V_2$, for the two mirror trajectories depicted in Fig. 3, measuring the amount of correlations between the early and late radiation. MI returns to its original value for trajectory $g_{\mathrm{II}}$, whereas it does not for $g_{\mathrm{I}}$. The loss of correlations $\mathrm{\Delta }I$ given in (34) serves to quantify unitarity violation. Here $a_1=0$, $b_1=1$, $a_2=3$, $b_2={10}^6$ with $\beta =7$ in (30).