Geometric interpretation of the triality relation in bipartite quantum systems

Abstract

We investigate quantum correlations and coherence in bipartite systems using the universal triality relation that connects coherence, predictability, and entanglement. The geometric structure underlying entanglement in bipartite two-qubit and two-qutrit systems is analyzed using wedge products of post-measurement vectors. For two-qubit states, concurrence emerges naturally as the area spanned by the post-measurement vectors, while in qutrit systems the generalized I-concurrence is associated with higher-dimensional geometric structures formed by multiple wedge products. We further analyze the relationship between wedge products, eigenvalues, and eigenvectors of correlated systems and discuss the universality of the triality relation in arbitrary dimensions. Our results provide a geometric interpretation of entanglement in d-dimensional cases and reveal a hierarchical structure connecting low- and high-dimensional quantum systems.