Geometric Entanglement Entropy on Projective Hilbert Space

• URL: http://arxiv.org/abs/2511.21186v1
• Date: Wed, 26 Nov 2025 09:03:20 GMT
• Title: Geometric Entanglement Entropy on Projective Hilbert Space
• Authors: Loris Di Cairano,
• Abstract summary: Entanglement for pure bipartite states is most commonly quantified in a state-by-state manner.<n>This provides a precise local characterization of how entangled a given state is.<n>In this work, we develop a geometric framework in which these questions become natural.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Entanglement for pure bipartite states is most commonly quantified in a state-by-state manner to each pure state of a bipartite system a scalar quantity, 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? 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-1/2 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.

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