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Physica Scripta

Geometric interpretation of the triality relation in bipartite quantum systems

Pratidhwani Swain, Ramita Sarkar, Sukanta K. Tripathy, Prasanta Kumar Panigrahi

Published Sep 7, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/aea381

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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.
Abstract The wedge product of post-measurement vectors of a two-qubit state gives rise to a parallelogram, whose ‘area’ has been shown to be equivalent to the generalized I-concurrence measure of entanglement. In multi-qudit systems, the wedge product of post-measurement vectors naturally leads to a higher dimensional parallelepiped which yields a modified faithful entanglement measure. Our new measure fine grains the entanglement monotone, wherein different entangled classes manifest with different geometries. We present a complete analysis of the bipartite qutrit case considering all possible geometric structures where three entanglement classes of pure bipartite qutrit states can be identified with different geometries of post-measurement vectors: three planar vectors; three mutually orthogonal vectors; and three vectors that are neither planar and not all of them are mutually orthogonal. It is further demonstrated that the geometric condition of area and volume maximization naturally leads to the maximization of entanglement. The wedge product approach uncovers an inherent geometry of entanglement and is found to be very useful for the characterization and quantification of entanglement in higher dimensional systems.
We investigate the entanglement structure of bipartite two-qutrit pure states from both geometric and operational perspectives.Using the eigenvalues of the reduced density matrix, we analyze how symmetric polynomials characterize pairwise and genuinely three-level correlations. We show that the determinant of the coefficient matrix defines a natural, rank-sensitive geometric invariant that vanishes for all rank-2 states and is nonzero only for rank-3 entangled states. An explicit analytic constraint relating this determinant-based invariant to the I-concurrence is derived, thereby defining the physically accessible region of two-qutrit states in invariant space. Furthermore, we establish an operational correspondence with three-path optical interferometry and analyze conditional visibility and predictability in a qutrit quantum erasure protocol, including the effects of unequal path transmittances. Numerical demonstrations confirm the analytic results and the associated complementarity relations. These findings provide a unified geometric and operational framework for understanding two-qutrit entanglement.
In quantum resource theories (QRTs), evidence of intrinsic connections between coherence and quantum correlations, including Bell nonlocality, entanglement, quantum steering, and so on, exists. However, building these relationships is a vital yet challenging task in multipartite quantum systems. Here, we focus on a unified framework of interpreting the interconversions among coherence and different quantum correlations in tripartite systems. In particular, an exact relation between the generalized geometric measure and the genuinely multipartite concurrence is derived for tripartite entanglement states. Then we obtain the trade-off relation between the first-order coherence and the genuine tripartite entanglement by the genuinely multipartite concurrence and concurrence fill. Furthermore, the trade-off relation between the maximum steering inequality violation and concurrence fill for arbitrary three-qubit states is found. In addition, we investigate the close relation between the maximum steering inequality violation and the first-order coherence. The results show that coherence and quantum correlations are intrinsically related and can be converted to one another in the framework of QRTs.
Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini--Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.
There are two important paradigms for defining quantum correlations in quantum information theory, viz. the information-theoretic and the entanglement-separability ones. We find an analytical relation between two measures of quantum correlations, one in each paradigm, and show that only a certain cone-like region on the two-dimensional space spanned by these measures is accessible to pure three-qubit states. The information-theoretic multiparty quantum correlation measure is related to the monogamy considerations of a bipartite information-theoretic quantum correlation measure, while the entanglement-separability multiparty measure is the generalized geometric measure, a genuine multiparty entanglement measure. We also find an analytical relation between two multiparty entanglement measures, and again obtain a cone-like accessible region in this case. One of the multisite measures in this case is related to the monogamy of a bipartite entanglement measure, while the other is again the generalized geometric measure.
Composite quantum systems can be in generic states characterized not only by entanglement but also by more general quantum correlations. The interplay between these two signatures of nonclassicality is still not completely understood. In this work we investigate this issue, focusing on computable and observable measures of such correlations: entanglement is quantified by the negativity N\mathcal{N}N, while general quantum correlations are measured by the (normalized) geometric quantum discord DG{D}_{G}DG​. For two-qubit systems, we find that the geometric discord reduces to the squared negativity on pure states, while the relationship ${D}_{G}\ensuremath{\ge}{\mathcal{N}}^{2}$ holds for arbitrary mixed states. The latter result is rigorously extended to pure, Werner, and isotropic states of two-qudit systems for arbitrary ddd, and numerical evidence of its validity for arbitrary states of a qubit and a qutrit is provided as well. Our results establish an interesting hierarchy, which we conjecture to be universal, between two relevant and experimentally friendly nonclassicality indicators. This ties in with the intuition that general quantum correlations should at least contain and in general exceed entanglement on mixed states of composite quantum systems.