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

Physics Department, College of Sciences, Shiraz University - Shiraz 71454, Iran

Received: 7 July 2025
Accepted: 29 June 2026

Abstract

Toric Code, a cornerstone of the topological order, admits numerous realizations on different planar graphs. We prove that all these realizations share a single, universal entanglement structure in a sense that any Toric Code state can be generated by applying a fixed set of non-local unitaries between similar quasi-one-dimensional quantum ladders. In particular, 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. This exact and graph-independent construction reveals that the long-range entanglement characterizing the Toric Code's topological phase has a precise decomposable structure in a sense that it arises entirely from a specific pattern of entanglement between lower-dimensional subsystems. This structural insight opens new avenues for classification and simulation of topological orders.