# Topological order on the Bloch sphere

## TL;DR

This paper constructs a Bloch sphere for a specific quantum system related to Kitaev's toric code, analyzes its topological order, and relates the findings to quantum search algorithms, revealing unique entanglement and separability properties.

## Contribution

It introduces a Bloch sphere for the toric code ground state, characterizes its topological order, and connects the structure to Grover rotations with gauge symmetry.

## Key findings

- Only one separable state on the sphere.
- The toric code ground state exhibits non-trivial topological order.
- Most states are neither trivial nor topologically ordered.

## Abstract

A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [M. Boyer, R. Liss, T. Mor, PRA 95, 032308 (2017)]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev's toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch sphere are neither ``trivial'' states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.