Geometric entanglement entropy on projective Hilbert space
Phys. Rev. A 113, 042448 – Published 20 April, 2026
DOI: https://doi.org/10.1103/3y4g-2ykl
Abstract
Entanglement for pure bipartite states is most commonly quantified on a state-by-state basis by assigning to each pure state a scalar value, such as the von Neumann entropy of a reduced density matrix. This provides a precise local characterization of “how entangled” a given state is. At the same time, this local description naturally invites a set of complementary, more global questions about the structure of the space of pure states: How abundant are the states with a given amount of entanglement within the full state space? Do the manifolds of constant entanglement exhibit distinct geometric regimes or “phases” when viewed in the natural geometry of projective Hilbert space? These questions shift the focus from assigning an entanglement value to a single state to understanding the global organization and geometry of entanglement across the entire manifold of pure states. In this work, we develop a geometric framework in which these questions become natural. We regard the projective Hilbert space of pure states, endowed with the Fubini-Study metric, as a Riemannian manifold and promote bipartite entanglement to a macroscopic functional on this manifold. Its level sets stratify the space of pure states into hypersurfaces of constant entanglement, and we define a geometric entanglement entropy as the log-volume of these hypersurfaces, weighted by the Fubini-Study gradient of entanglement. This quantity plays the role of a microcanonical entropy in “entanglement space”: it measures the degeneracy of a given entanglement value in the natural quantum geometry. The framework is illustrated first in the simplest case of a single spin- (where the Fubini-Study metric reduces to the Bloch sphere metric), and then for bipartite entanglement of spin systems, including a two-qubit example where explicit calculations can be carried out, along with a sketch of the extension to spin chains.
