Maximally Entangled Two-Qutrit Quantum Information States and De Gua's Theorem for Tetrahedron

TL;DR

This paper explores geometric relations in entangled two-qubit and two-qutrit states, revealing novel connections between entanglement measures and classical geometric theorems like De Gua's theorem.

Contribution

It introduces new geometric interpretations of entanglement measures for two-qubit and two-qutrit states, including a relation involving De Gua's theorem for maximally entangled two-retrit states.

Findings

Concurrence relates to the double area of a parallelogram for rebit states.

A Pythagoras-type relation for the reduced density matrix of two-qutrit states.

Maximally entangled two-retrit states satisfy De Gua's theorem.

Abstract

Geometric relations between separable and entangled two-qubit and two-qutrit quantum information states are studied. To characterize entanglement of two qubit states, we establish a relation between reduced density matrix and the concurrence. For the rebit states, the geometrical meaning of concurrence as double area of a parallelogram is found and for generic qubit states it is expressed by determinant of the complex Hermitian inner product metric, where reduced density matrix coincides with the inner product metric. In the case of generic two-qutrit state, for reduced density matrix we find Pythagoras type relation, where the concurrence is expressed by sum of all $2 \times 2$ minors of $3\times3$ complex matrix. For maximally entangled two-retrit state, this relation is just De Gua's theorem or a three-dimensional analog of the Pythagorean theorem for triorthogonal tetrahedron areas. Generalizations of our results for arbitrary two-qudit states are discussed