[Paper Review] Effective reflected entropy and entanglement negativity for general 2D eternal black holes

Jiong Lin, Yizhou Lu|arXiv (Cornell University)|Apr 18, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper develops a holographic framework to compute effective reflected entropy and entanglement negativity for general 2D eternal black holes coupled to CFT matter in the large $c$ limit using the replica trick. It shows that both quantities are encoded in the areas of modified backreacting cosmic branes in $<mathrm>{AdS}_3$ bulk, and confirms consistency between two island formula proposals for entanglement negativity, with reflected entropy saturating bounds analogous to GHZ states.

ABSTRACT

Both reflected entropy and entanglement negativity provide valid measures of entanglement between subsystems of a mixed state. For general 2D eternal black holes coupled with CFT matters in large $c$ limit, we perform the replica-trick computation and find that both effective Renyi reflected entropy and effective entanglement negativity can be expressed in terms of the combination of modified backreacting cosmic branes in $\mathrm{AdS}_3$ bulk. We then develop a holographic scheme to calculate effective reflected entropy and entanglement negativity for general 2D eternal black holes coupled with CFT matters in large $c$ limit. Using the scheme, we check the consistency condition of the island formulae for entanglement negativity and reflected entropy. We find that the combinations of modified backreacting cosmic branes in $\mathrm{AdS}_3$ bulk from the two island proposals of entanglement negativity exactly match with each other. Finally, we study the saturation of the reflected entropy inequality.

Motivation & Objective

• To develop a holographic scheme for computing effective Renyi reflected entropy and entanglement negativity in 2D eternal black holes coupled to CFT matter.
• To verify the consistency of two distinct island formula proposals for entanglement negativity in the context of general 2D eternal black holes.
• To investigate the saturation conditions of the reflected entropy inequality and its physical implications in the large $c$ limit.
• To clarify the role of modified backreacting cosmic branes in encoding entanglement measures in the $<mathrm>{AdS}_3$ bulk.

Proposed method

• Performing replica-trick computations in the large $c$ limit to derive effective Renyi reflected entropy and entanglement negativity.
• Mapping the results to the areas of modified backreacting cosmic branes in $<mathrm>{AdS}_3$ bulk, with specific contributions from twist operators of various scaling dimensions.
• Applying the minimal QEWCS proposal and the 1/2-order Renyi entropy approach to compute entanglement negativity and cross-check consistency.
• Using the holographic scheme to evaluate reflected entropy and test the Araki-Lieb inequality saturation in the context of black hole and radiation subsystems.
• Identifying which cosmic brane configurations contribute to the final expressions based on scaling dimensions of twist operators.
• Deriving expressions for mutual information and comparing them with reflected entropy to verify the inequality bounds.

Experimental results

Research questions

• RQ1 How can effective reflected entropy and entanglement negativity be computed holographically for general 2D eternal black holes in the large $c$ limit?
• RQ2 Do the two distinct island formula proposals for entanglement negativity yield consistent results in this setup?
• RQ3 Under what conditions does the reflected entropy inequality saturate, and what does this imply about the underlying quantum state?
• RQ4 Which configurations of modified backreacting cosmic branes contribute to the final expressions for reflected entropy and entanglement negativity?
• RQ5 How do the scaling dimensions of twist operators influence the contributions of cosmic branes to the entanglement measures?

Key findings

• Effective Renyi reflected entropy is determined by cosmic branes connecting twist operators with scaling dimensions $h_{g_A g_B^{-1}}$, $h_{g_B g_A^{-1}}$, and $h_{g_A g_B^{-1}}$, $h_{g_A^{-1}}$, $h_{g_B}$, while terms with $h_{g_A}$ and $h_m$ cancel out.
• Effective entanglement negativity arises solely from cosmic branes involving twist operators of scaling dimensions $h_{\tau^2_{n_e}}$, $h_{\bar{\tau}^2_{n_e}}$, and $h_{\tau_{n_e}}$, $h_{\tau_{n_e}}$, $h_{\bar{\tau}^2_{n_e}}$, as the $\tau_{n_e}$-related terms vanish when $n_e \to 1$.
• The two island formula proposals for entanglement negativity—minimal QEWCS and 1/2-order Renyi entropy—yield identical results, confirming consistency for general 2D eternal black holes.
• The islands for reflected entropy and entanglement negativity coincide, implying a unified geometric structure in the bulk for both measures.
• Reflected entropy between the black hole and radiation saturates the upper bound of the inequality $I(A:B) \leq S_R(A:B) \leq 2\min\{S(A),S(B)\}$, consistent with Araki-Lieb inequality saturation.
• Reflected entropy between two radiation regions saturates only the lower bound, analogous to a GHZ state, indicating a tripartite entanglement structure.