Universal representation of the long-range entanglement in the family of Toric Code states

arXiv:2509.03422 · quant-ph, cond-mat.str-el · Submitted 2025-09-03 · Read on arXiv

quant-ph, cond-mat.str-el

Submitted: 2025-09-03

Updated: 2025-09-03

Comments: Submitted to Europhysics Letters (EPL), 7 pages, 4 figures

Journal ref: Europhysics Letters, 155 58001 (2026)

License: http://creativecommons.org/licenses/by-nc-sa/4.0/

The gist: Since the long range entanglement is a universal characteristic of topological quantum states belonging to the same class, a suitable mathematical representation of the long range entanglement has to

Terminology

Entanglement — 64× in this paper · explained in 3 episodes
Topological Order — 17× in this paper · explained in 2 episodes
Toric Code — 16× in this paper
Area Law — 3× in this paper · explained in 1 episode
Entanglement Entropy — 3× in this paper · explained in 2 episodes
Phase Transitions — 3× in this paper · explained in 2 episodes
Anyon Condensation — 2× in this paper · explained in 1 episode
Degeneracy — 2× in this paper · explained in 3 episodes
Multipartite Entanglement — 2× in this paper · explained in 2 episodes
Symmetry Breaking — 2× in this paper · explained in 3 episodes
Anyons — 1× in this paper · explained in 2 episodes
Decoupling — 1× in this paper · explained in 2 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Trace — 1× in this paper · explained in 2 episodes
Trace Distance — 1× in this paper · explained in 3 episodes

Abstract

Since the long range entanglement is a universal characteristic of topological quantum states belonging to the same class, a suitable mathematical representation of the long range entanglement has to be also universal. In this Letter, we introduce such a representation for the family of Toric Code states by using Kitaev's Ladders as building blocks. We consider Toric Code states corresponding to various planar graphs and apply non-local dientanglers to qubits corresponding to non-contractible cycles that satisfy a topological constraint. We demonstrate that, independent of the geometry of the underlying graph, disentanglers convert Toric Code states into a tensor product of Kitaev's Ladder states. Since Kitaev's Ladders with arbitrary geometric configurations include the short-range entanglements, we conclude that the above universal and non-local pattern of entanglement between ladders is responsible of the long-range entanglement inherent in Toric Code states. Our result emphasizes in the capability of such non-local representations to describe topological order in ground-state wave functions of topological quantum systems.

Terminology

Abstract

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