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Entanglement negativity in decohered topological states

Kang-Le Cai, Meng Cheng

Abstract

We investigate universal entanglement signatures of mixed-state phases obtained by decohering pure-state topological order (TO), focusing on topological corrections to logarithmic entanglement negativity and mutual information: topological entanglement negativity (TEN) and topological mutual information (TMI). For Abelian TOs under decoherence, we develop a replica field-theory framework based on a doubled-state construction that relates TEN and TMI to the quantum dimensions of domain-wall defects between decoherence-induced topological boundary conditions, yielding general expressions in the strong-decoherence regime. We further compute TEN and TMI exactly for decohered $G$-graded string-net states, including cases with non-Abelian anyons. We interpret the results within the strong one-form-symmetry framework for mixed-state TOs: TMI probes the total quantum dimension of the emergent premodular anyon theory, whereas TEN detects only its modular part.

Entanglement negativity in decohered topological states

Abstract

We investigate universal entanglement signatures of mixed-state phases obtained by decohering pure-state topological order (TO), focusing on topological corrections to logarithmic entanglement negativity and mutual information: topological entanglement negativity (TEN) and topological mutual information (TMI). For Abelian TOs under decoherence, we develop a replica field-theory framework based on a doubled-state construction that relates TEN and TMI to the quantum dimensions of domain-wall defects between decoherence-induced topological boundary conditions, yielding general expressions in the strong-decoherence regime. We further compute TEN and TMI exactly for decohered -graded string-net states, including cases with non-Abelian anyons. We interpret the results within the strong one-form-symmetry framework for mixed-state TOs: TMI probes the total quantum dimension of the emergent premodular anyon theory, whereas TEN detects only its modular part.
Paper Structure (20 sections, 105 equations, 5 figures)

This paper contains 20 sections, 105 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Entanglement measures in mixed states
  3. Field-theoretic calculation for decohered Abelian TO
  4. Topological entanglement negativity
  5. Topological mutual information
  6. Pure state TO
  7. Decohered TO
  8. Example: decohered toric code phase
  9. TEN and TMI of decohered G-graded string-net states
  10. Decohered doubled Ising string-net state
  11. Decohered G-graded string-net states
  12. One-form symmetry of mixed-state TO
  13. Conclusions and discussions
  14. Acknowledgment
  15. Representation theory of string operator algebra
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Levin--Wen geometry for extracting the topological correction in $S_A$. The regions $A$, $B$, and $C$ are chosen such that the volume-law and area-law contributions cancel out in Eq. \ref{['Eq_EM_CMI']}.
  • Figure 2: Path-integral representation of Eq. \ref{['Eq_TQFT_Doubled']}. Without decoherence, the reference bra states $\langle\!\bra{\mathcal{E}_n}$ and $\langle\!\bra{\mathcal{E}_n} (R_A^*)^2$ impose corresponding boundary conditions at the $\tau = 0$ surface. The effect of decoherence can be captured by interactions (represented by the grey region) that couple $\phi_i$ and $\bar{\phi}_i$ at $\tau=0$.
  • Figure 3: Schematic picture in which the $2n$ replicas are compressed into a single piece. The red dots denote the domain walls between two boundary conditions on a spatial slice.
  • Figure 4: Configuration with four junctions between $\mathcal{A}_0'$ and $\mathcal{A}_1'$.
  • Figure 5: Reduced tree-like string-net configuration within the subregion $A$. The edges $q_i$ are on the boundary of $A$, while the edges $s_i$ are internal to $A$.