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Entanglement entropy and negative energy in two dimensions

Eugenio Bianchi, Matteo Smerlak

TL;DR

This work links transient negative energy densities in two-dimensional conformal field theories to the entanglement structure of the vacuum by deriving an exact energy–entropy relation at future null infinity. The authors express the renormalized entanglement entropy S(u) in terms of the peeling function κ via S(u) = (c/12) ∫ κ, and relate the outgoing flux to S through F(u) = (1/2π)[(6/c)(dS/du)^2 + d^2S/du^2], revealing that any nontrivial conformal vacuum must emit negative energy in some interval. Recasting the problem as a zero-energy Schrödinger scattering with V = (12π/c)F leads to a resonance constraint that enforces a global condition ∫ F e^{6S/c} du = 0 and implies a formal integral series for S(u) in terms of F. Applied to unitary black hole evaporation, the framework predicts non-monotonic mass loss near the Page time, i.e., a brief transient increase in the retarded mass, illustrating how unitarity constrains semiclassical evaporation. The results extend to moving mirrors and squeezed states, underscoring a deep link between vacuum entanglement and energy flux in 2D CFTs.

Abstract

It is well known that quantum effects can produce negative energy densities, though for limited times. Here we show in the context of two-dimensional CFT that such negative energy densities are present in any non-trivial conformal vacuum and can be interpreted in terms of the entanglement structure of such states. We derive an exact identity relating the outgoing energy flux and the entanglement entropy in the in-vacuum. When applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss.

Entanglement entropy and negative energy in two dimensions

TL;DR

This work links transient negative energy densities in two-dimensional conformal field theories to the entanglement structure of the vacuum by deriving an exact energy–entropy relation at future null infinity. The authors express the renormalized entanglement entropy S(u) in terms of the peeling function κ via S(u) = (c/12) ∫ κ, and relate the outgoing flux to S through F(u) = (1/2π)[(6/c)(dS/du)^2 + d^2S/du^2], revealing that any nontrivial conformal vacuum must emit negative energy in some interval. Recasting the problem as a zero-energy Schrödinger scattering with V = (12π/c)F leads to a resonance constraint that enforces a global condition ∫ F e^{6S/c} du = 0 and implies a formal integral series for S(u) in terms of F. Applied to unitary black hole evaporation, the framework predicts non-monotonic mass loss near the Page time, i.e., a brief transient increase in the retarded mass, illustrating how unitarity constrains semiclassical evaporation. The results extend to moving mirrors and squeezed states, underscoring a deep link between vacuum entanglement and energy flux in 2D CFTs.

Abstract

It is well known that quantum effects can produce negative energy densities, though for limited times. Here we show in the context of two-dimensional CFT that such negative energy densities are present in any non-trivial conformal vacuum and can be interpreted in terms of the entanglement structure of such states. We derive an exact identity relating the outgoing energy flux and the entanglement entropy in the in-vacuum. When applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss.

Paper Structure

This paper contains 8 sections, 21 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Conformal vacuum.
  3. Renormalized entanglement entropy.
  4. Energy-entropy relation at future null infinity.
  5. Mapping to a scattering problem.
  6. Application to unitary black hole evaporation.
  7. Conclusion.
  8. Acknowledgements.

Figures (2)

  • Figure 1: Conformal diagram of an asymptotically flat half-plane with a reflecting boundary, defining a canonical mapping $v=p(u)$ between future null infinity $\mathcal{I}^{+}$ and past null infinity $\mathcal{I}^{-}$.
  • Figure 2: Left: Entanglement entropy in black hole evaporation, as posited by Page Page:1993bv. Right: The corresponding flux function (normalized to the Hawking flux $F_{H} \sim\hbar/M_{\textrm{ADM}}^{2}$), as derived from \ref{['riccati']}. (The Hawking thermal entropy and mass law (dashed line) are for reference, and $\tau\sim M_{\textrm{ADM}}^{3}/\hbar$ denotes the Hawking evaporation time.)