[Paper Review] Upgrading the Bloch Sphere: Projective Space Foliated by Klein Bottles as a Geometrical Representation of Two-Qubit States and Their Entanglement

Oscar Perdómo, Vicente Leyton‐Ortega+1 more|arXiv (Cornell University)|Mar 5, 2019

Fractal and DNA sequence analysisBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper proposes a novel geometric representation of two-qubit pure states with real amplitudes using real projective space, where maximally entangled states form two orthogonal circles and unentangled states are realized as a Klein bottle at distance $π/4$. The distance from maximally entangled states determines entanglement entropy via a derived formula, and all such states are connected via local gates and one controlled-Z gate, revealing a foliation of projective space by Klein bottles parametrized by entanglement distance.

ABSTRACT

We study the set of two-qubit pure states with real amplitudes and their geometrical representation onto a real projective space. 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 this space, 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 Klein bottle. Finally, for every $0< d < \pi/4$ the point $d$ corresponds to a disjoint pair of Klein bottles. 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 to clearly see the known result 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.

Motivation & Objective

• To develop a geometric representation of two-qubit pure states with real amplitudes in real projective space.
• To characterize the distribution of maximally entangled states and their relation to unentangled (product) states in this geometry.
• To establish a distance-based classification of states under local unitary equivalence, parameterized by the interval $[0, \pi/4]$.
• To derive a closed-form expression for entanglement entropy as a function of geometric distance from maximal entanglement.
• To demonstrate that any two such states can be connected using only single-qubit gates and one controlled-Z gate, leveraging the geometric structure.

Proposed method

• Represent the set of two-qubit pure states with real amplitudes as points in real projective space $\mathbb{RP}^3$.
• Define a Riemannian metric on $\mathbb{RP}^3$ to measure distances between states under local unitary transformations.
• Identify the maximally entangled states (locally equivalent to Bell states) as two disjoint, perpendicular circles in this space.
• Show that states at distance $d$ from the maximally entangled circles form a pair of disjoint Klein bottles for $0 < d < \pi/4$, with $d = \pi/4$ corresponding to unentangled states.
• Derive the entanglement entropy $S(d) = 1 - \log_2 \sqrt{\frac{(1+\sin 2d)^{1+\sin 2d}}{(1-\sin 2d)^{-1+\sin 2d}}}$ as a function of geometric distance $d$.
• Use the geometric structure to prove that any two two-qubit states with real amplitudes are connected by a circuit with only single-qubit gates and one controlled-Z gate.

Experimental results

Research questions

• RQ1 How can two-qubit pure states with real amplitudes be geometrically represented in a way that captures their entanglement structure?
• RQ2 What is the geometric location of maximally entangled states and unentangled states in this representation?
• RQ3 How does the distance from the maximally entangled states relate to the entanglement entropy of a state?
• RQ4 Can the set of all such states be foliated by topological objects like Klein bottles, and what is their geometric significance?
• RQ5 What is the minimal quantum circuit structure required to connect any two such states, and how does geometry inform this?

Key findings

• The set of maximally entangled states forms two disjoint, perpendicular circles in the real projective space representation of two-qubit pure states with real amplitudes.
• Unentangled (product) states are located exactly $\pi/4$ units away from the maximally entangled states in the Riemannian metric, and geometrically form a Klein bottle.
• For each $d \in (0, \pi/4)$, the set of states at distance $d$ from the maximally entangled states forms a pair of disjoint Klein bottles.
• The entanglement entropy of a state at distance $d$ from maximal entanglement is given by $S(d) = 1 - \log_2 \sqrt{\frac{(1+\sin 2d)^{1+\sin 2d}}{(1-\sin 2d)^{-1+\sin 2d}}}$.
• All two-qubit states with real amplitudes are connected by circuits composed of only single-qubit gates and one controlled-Z gate, as confirmed by the geometric foliation.
• The space of states is foliated by Klein bottles parameterized by the distance $d \in [0, \pi/4]$, with $d=0$ corresponding to the maximally entangled states and $d=\pi/4$ to the unentangled states.