Entanglement Suppression Due to Black Hole Scattering

We consider the evolution of entanglement entropy in a two-dimensional conformal field theory with a holographic dual. Specifically, we are interested in a class of excited states produced by a combination of pure-state (local operator) and mixed-state local quenches. We employ a method that allows us to determine the full time evolution analytically. While a single insertion of a local operator gives rise to a logarithmic time profile of entanglement entropy relative to the vacuum, we find that this growth is heavily suppressed in the presence of a mixed-state quench, reducing it to a time-independent constant bump. The degree of suppression depends on the relative position of the quenches as well as the ratio of regularization parameters associated with the quenches. This work sheds light on the interesting properties of gravitational scattering involving black holes.

💡 Research Summary

This paper investigates the time evolution of entanglement entropy (EE) in a two‑dimensional conformal field theory (CFT) that possesses a holographic dual, focusing on a novel class of excited states generated by the simultaneous application of two distinct local quenches. The first quench is a pure‑state local operator quench: a heavy primary operator ( \mathcal{O} ) (with conformal dimension ( h\sim \mathcal{O}(c) )) is inserted at the origin and regularized by a UV cutoff ( \delta ). The second quench is a mixed‑state local quench, defined by a canonical ensemble of all local primary operators and their descendants, which in the bulk corresponds to a localized black‑hole geometry. By performing the Euclidean path integral on a plane that has been deformed by the mixed‑state quench and then inserting the pure‑state operator, the authors obtain the full replica‑trick expression for the ( n )-th Rényi entropy.

The analysis relies on the heavy‑heavy‑light‑light (HHLL) approximation of conformal blocks. In the holographic regime the four‑point function is dominated by the identity block, allowing the authors to replace the full block expansion with a single HHLL block. Taking the ( n\to1 ) limit (where the twist operator dimension becomes light) yields an analytic expression for the entanglement entropy shift ( \Delta S_A ) in terms of the cross‑ratios ( z,\bar z ). The cross‑ratios are expressed as functions of the subsystem endpoints ( a,b ), the insertion time ( t ), and the regularization parameters ( \delta ) (for the pure quench) and ( s ) (the spatial width associated with the mixed quench).

When only the pure‑state operator is present, the well‑known result is recovered: for a large interval and intermediate times the EE grows logarithmically, ( \Delta S_A \sim \frac{c}{6}\log t ), with a coefficient that depends on the heavy‑operator parameter ( \alpha_H=\sqrt{1-24h_{\mathcal O}/c} ). The authors carefully discuss the branch‑choice for the phases of the cross‑ratios, introducing two “channels” labelled by integers ( (m,\bar m) ). The physical prescription is to select, at each time, the channel that minimizes the entropy, reproducing the quasiparticle picture in which one member of the entangled pair enters the interval while the other remains outside.

The central new result concerns the effect of the mixed‑state quench. Two regimes are examined:

1. ( \delta \ll s ) (the mixed quench is broad compared with the UV regulator). In this limit the cross‑ratios become almost real and the HHLL block reduces to a constant. Consequently the logarithmic growth disappears entirely; the EE shift becomes a time‑independent “bump” of magnitude ( \Delta S_{\text{const}} \sim \frac{c}{6}\log!\bigl(1+\alpha_H s/\delta\bigr) ). This demonstrates a complete suppression of entanglement growth.

2. ( s \ll \delta ) (the mixed quench is sharply localized). Here the logarithmic term survives but is multiplied by a small factor ( \alpha_H (s/\delta) \ll 1 ). The EE behaves as ( \Delta S_A \approx \frac{c}{6},\alpha_H\frac{s}{\delta},\log t ), i.e. the growth is heavily attenuated.

In both regimes the suppression depends sensitively on the relative position of the two quenches (the distance between the operator insertion point and the interval endpoints) and on the ratio ( s/\delta ). The authors also verify that the minimal‑channel prescription continues to hold when the mixed quench is present.

From the holographic perspective, the mixed‑state quench creates a localized black hole with temperature and entropy determined by ( s ) and ( \delta ). The pure‑state operator corresponds to a heavy particle falling into the AdS bulk. The combined state therefore describes a heavy particle scattering off a localized black hole. The suppression of EE growth is interpreted as the particle’s information being partially absorbed by the black hole, reducing the amount of entanglement that can be shared with the exterior region. This provides a concrete field‑theoretic illustration of information‑loss‑type phenomena in a controlled setting.

The paper also includes two appendices. Appendix A analyses the same setup in the thermal AdS phase (i.e., when the bulk geometry does not contain a black hole), confirming that the suppression mechanism is absent in that phase. Appendix B extends the discussion to include a chemical potential, showing that the qualitative picture remains unchanged when a conserved charge is present.

Overall, the work offers a detailed, analytically tractable study of how mixed‑state local quenches can dramatically alter entanglement dynamics in holographic CFTs. By bridging the quasiparticle picture, conformal block techniques, and bulk gravitational intuition, it deepens our understanding of the interplay between quantum information and black‑hole physics.