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Relative Rényi Entropy Under Local Quenches in 2D CFTs

Zi-Xuan Zhao, Song He, Hao Ouyang, Hong-an Zeng, Yu-Xuan Zhang

TL;DR

The paper investigates relative Rényi entropy under local quenches in 2D CFTs, focusing on RCFTs and holographic CFTs. It develops a replica-based framework to compute the $k$-th RRE between various excited states, revealing monotonic time evolution for descendants in RCFTs and a deep link to holomorphic data, while establishing a holographic connection where collision-relative entropy recovers entanglement-wedge geometry via a Hellinger-type metric. A finite-dimensional matrix structure is shown to govern RRE for linear combinations of operators, and a quasi-particle interpretation explains cases where RRE fails to distinguish certain states. The results contribute to understanding how quantum information measures encode both boundary state distinguishability and bulk reconstruction in AdS/CFT, and they open questions about symmetry conditions and analytic continuation in these settings.

Abstract

We study the relative Rényi entropy (RRE) under local quenches in two-dimensional conformal field theories (CFTs), focusing on rational CFTs (RCFTs) and holographic CFTs. In RCFTs, the RRE evolves as a monotonic function over time, depending on finite-dimensional matrices. It is sometimes symmetric, prompting an exploration of its relation to the trace squared distance. We also observe that relative entropy can fail to distinguish between operators, as it only captures information entering/exiting the subsystem. In holographic CFTs, an analytic continuation of the RRE reveals insights into the entanglement wedge, offering a new perspective on bulk geometry in AdS/CFT. Our results deepen the understanding of quantum information measures in RCFTs and holographic CFTs, highlighting connections to distinguishability and bulk reconstruction.

Relative Rényi Entropy Under Local Quenches in 2D CFTs

TL;DR

The paper investigates relative Rényi entropy under local quenches in 2D CFTs, focusing on RCFTs and holographic CFTs. It develops a replica-based framework to compute the -th RRE between various excited states, revealing monotonic time evolution for descendants in RCFTs and a deep link to holomorphic data, while establishing a holographic connection where collision-relative entropy recovers entanglement-wedge geometry via a Hellinger-type metric. A finite-dimensional matrix structure is shown to govern RRE for linear combinations of operators, and a quasi-particle interpretation explains cases where RRE fails to distinguish certain states. The results contribute to understanding how quantum information measures encode both boundary state distinguishability and bulk reconstruction in AdS/CFT, and they open questions about symmetry conditions and analytic continuation in these settings.

Abstract

We study the relative Rényi entropy (RRE) under local quenches in two-dimensional conformal field theories (CFTs), focusing on rational CFTs (RCFTs) and holographic CFTs. In RCFTs, the RRE evolves as a monotonic function over time, depending on finite-dimensional matrices. It is sometimes symmetric, prompting an exploration of its relation to the trace squared distance. We also observe that relative entropy can fail to distinguish between operators, as it only captures information entering/exiting the subsystem. In holographic CFTs, an analytic continuation of the RRE reveals insights into the entanglement wedge, offering a new perspective on bulk geometry in AdS/CFT. Our results deepen the understanding of quantum information measures in RCFTs and holographic CFTs, highlighting connections to distinguishability and bulk reconstruction.

Paper Structure

This paper contains 18 sections, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup in 2D CFTs
  3. Relative Rényi entropy for different states
  4. $k^{\rm th}$ relative Rényi entropy for $|\Omega\rangle$ and $\mathcal{O}(x)|\Omega\rangle$
  5. $k^{\rm th}$ relative Rényi entropy for $L_{-n}\mathcal{O}(x)|\Omega\rangle$ and $\mathcal{O}(x)|\Omega\rangle$
  6. $k^{\rm th}$ relative Rényi entropy for two general descendant states
  7. $k^{\rm th}$ relative Rényi entropy and trace square distance
  8. Relative Rényi entropy for linear combinations of operators
  9. Relative Rényi entropy and quasi-particle
  10. Relative Rényi entropy and entanglement wedge
  11. Conclusions and discussions
  12. Useful formulae for computing the k-th RRE
  13. Relative Rényi entropies for two locally excited states
  14. $|\Omega\rangle$ and $\mathcal{O}(x)|\Omega\rangle$
  15. $L_{-n}\mathcal{O}(x)|\Omega\rangle$ and $\mathcal{O}(x)|\Omega\rangle$
  16. ...and 3 more sections

Figures (2)

  • Figure 1: The quasi-particle moving to the left contains only the anti-holomorphic information of the operator, while the quasi-particle moving to the right contains only the holomorphic information of the operator.
  • Figure 2: We select subsystem $A$ as $(-\infty, 0]$, and the excitation point of the operator is on the right side of $A$.