Visualizing Two Qubits

TL;DR

This paper introduces a three-dimensional visualization of two-qubit states using SLOCC classes, providing geometric insights into entanglement, separability, and Bell inequalities, and demonstrating limitations of measurement strategies.

Contribution

It presents a novel geometric visualization method for two-qubit states that clarifies entanglement and Bell inequality properties, offering new proofs and insights.

Findings

Peres separability test is iff for two qubits

Bell inequalities can be visualized as geometric shapes

Three measurements do not reveal new entanglement classes

Abstract

The notions of entanglement witnesses, separable and entangled states for two qubits system can be visualized in three dimensions using the SLOCC equivalence classes. This visualization preserves the duality relations between the various sets and allows us to give ``proof by inspection'' of a non-elementary result of the Horodeckies that for two qubits, Peres separability test is iff. We then show that the CHSH Bell inequalities can be visualized as circles and cylinders in the same diagram. This allows us to give a geometric proof of yet another result of the Horodeckies, which optimizes the violation of the CHSH Bell inequality. Finally, we give numerical evidence that, remarkably, allowing Alice and Bob to use three rather than two measurements each, does not help them to distinguish any new entangled SLOCC equivalence class beyond the CHSH class.