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Deconstructing the Wormhole: Factorization, Entanglement and Decoherence

Herman Verlinde

TL;DR

This work elucidates how ensemble averaging in holography links the late-time universal behavior of the spectral form factor to the second Rényi entropy of a thermo-field double–like mixed state, using wormhole saddles to realize non-factorization and decoherence. It introduces a generalized thermo-mixed double (TMD) with old (fully dephased) and new (partially dephased) variants, decomposing the Hilbert space into classical, quantum, and uncorrelated sectors to capture competing bulk connectivity and boundary entanglement. The analysis extends to higher Rényi orders, predicting that connected wormholes correspond to n-fold trumpet geometries Σ_n, with the new TMD yielding tr(ρ_TMD^n) ∼ e^{-2(n−1)S₀} near the dip, in agreement with Z(Σ_n). By comparing von Neumann entropy and mutual information across factorized, TFD, and TMD states, the paper shows how entropy and correlations distribute among the three quantum-number sectors, and argues that typical two-sided black holes correspond to the old TMD while a code subspace may preserve entanglement, linking non-factorization, entanglement, and decoherence in holography.

Abstract

We study the role of ensemble averaging in holography by exploring the relation between the universal late time behavior of the spectral form factor and the second Renyi entropy of a thermal mixed state of the doubled system. Both quantities receive contributions from wormhole saddle-points: in the former case they lead to non-factorization while in the latter context they quantify decoherence due to the interaction with an environment. Our reasoning indicates that the space-time continuity responsible for non-factorization and space-time continuity through entanglement are in direct competition with each other. In an accompanying paper, we examine this dual relationship in a general class of 1D quantum systems with the help of a simple geometric path integral prescription.

Deconstructing the Wormhole: Factorization, Entanglement and Decoherence

TL;DR

This work elucidates how ensemble averaging in holography links the late-time universal behavior of the spectral form factor to the second Rényi entropy of a thermo-field double–like mixed state, using wormhole saddles to realize non-factorization and decoherence. It introduces a generalized thermo-mixed double (TMD) with old (fully dephased) and new (partially dephased) variants, decomposing the Hilbert space into classical, quantum, and uncorrelated sectors to capture competing bulk connectivity and boundary entanglement. The analysis extends to higher Rényi orders, predicting that connected wormholes correspond to n-fold trumpet geometries Σ_n, with the new TMD yielding tr(ρ_TMD^n) ∼ e^{-2(n−1)S₀} near the dip, in agreement with Z(Σ_n). By comparing von Neumann entropy and mutual information across factorized, TFD, and TMD states, the paper shows how entropy and correlations distribute among the three quantum-number sectors, and argues that typical two-sided black holes correspond to the old TMD while a code subspace may preserve entanglement, linking non-factorization, entanglement, and decoherence in holography.

Abstract

We study the role of ensemble averaging in holography by exploring the relation between the universal late time behavior of the spectral form factor and the second Renyi entropy of a thermal mixed state of the doubled system. Both quantities receive contributions from wormhole saddle-points: in the former case they lead to non-factorization while in the latter context they quantify decoherence due to the interaction with an environment. Our reasoning indicates that the space-time continuity responsible for non-factorization and space-time continuity through entanglement are in direct competition with each other. In an accompanying paper, we examine this dual relationship in a general class of 1D quantum systems with the help of a simple geometric path integral prescription.

Paper Structure

This paper contains 5 sections, 42 equations, 1 figure.

Table of Contents

  1. Introduction: factorization and purity
  2. Thermal mixed double, old and new
  3. Higher spectral form factor and Rényi entropy
  4. Von Neumann entropy and mutual information
  5. Conclusion: non-factorization and decoherence

Figures (1)

  • Figure 1: Depiction of the second Rényi entropy of the new thermo-mixed double state. The first factor in \ref{['tmdfacto']} describes the classically correlated sector: the quantum number $j$ is shared both between the $L$ and $R$ sectors (correlated) as well as between the ket and bra states (classical). This sector in non-factorized, both between the two ends of the wormhole (left and right in the figure) and between the $L$ and $R$ sector (front and back). The second factor in \ref{['tmdfacto']} describes the quantum entangled sector. This sector is in a pure state: the quantum number $m$ is shared coherently between the $L$ and $R$ sectors (entangled) but not across the ket and bra states (quantum). This sector in factorized between the two ends of the wormhole but the $L$ and $R$ sides (front and back) are connected. The third factor in \ref{['tmdfacto']} describes the classical uncorrelated sector: the $L$ and $R$ sectors have independent quantum numbers $s$ and $s'$ (uncorrelated) but the ket and bra states share the same quantum number (classical). This sector in non-factorized between the two ends of the wormhole but the $L$ and $R$ sector (front and back) are disconnected.