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

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