# Entanglement types for two-qubit states with real amplitudes

## TL;DR

This paper explores the geometric structure of two-qubit states with real amplitudes, revealing how entanglement classes form specific geometric shapes and how this understanding simplifies quantum circuit design.

## Contribution

It provides a novel geometric representation of real-amplitude two-qubit states, linking entanglement measures to distances on a sphere and tori, and shows how to connect states with simple circuits.

## Key findings

- Maximally entangled states form two disjoint circles on the sphere.
- Unentangled states are at a distance of π/4 from maximally entangled states.
- Any two real-amplitude two-qubit states can be connected with single-qubit gates and one controlled-Z gate.

## Abstract

We study the set of two-qubit pure states with real amplitudes and their geometrical representation in the three-dimensional sphere. In this representation, we show that the maximally entangled states --those locally equivalent to the Bell States --form two disjoint circles perpendicular to each other. We also show that taking the natural Riemannian metric on the sphere, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled, or so-called product states, are $\pi/4$ units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval $[0,\pi/4]$ with the point $0$ corresponding to the maximally entangled states. The point $\pi/4$ corresponds to the unentangled states which geometrically are described by a torus. Finally, for every $0< d < \pi/4$ the point $d$ corresponds to a disjoint pair of torus. We also show that if a state is $d$ units away from the maximally entangled states, then its entanglement entropy is $S(d) = 1- \log_2 \sqrt{\frac{(1+\sin 2 d)^{1+\sin 2 d}}{(1-\sin 2 d)^{-1+\sin 2 d}}}$. Finally, we also show how this geometrical interpretation allows us to clearly see that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate.