Geometric Entanglement Entropy on Projective Hilbert Space

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.

Figures (2)

• Figure 1: Geometric entropy $S_{\mathrm{geo}}(\theta)$ for a pair of qubits, as a function of the Schmidt angle $\theta$ in the parametrization $|\psi(\theta)\rangle=\cos\theta\,|00\rangle+\sin\theta\,|11\rangle$. The entropy diverges to $-\infty$ at $\theta\to 0,\pi/2$, corresponding to almost product states, showing that such low-entanglement states occupy a negligible Fubini-Study volume. The cusp at $\theta=\pi/4$ signals a strong concentration of volume around nearly maximally entangled states; the symmetry about $\pi/4$ reflects the degeneracy $\theta\leftrightarrow \tfrac{\pi}{2}-\theta$ of the Schmidt spectrum.
• Figure 2: Trace of the Weingarten operator, $\mathrm{Tr}\,W(\theta)=\mathrm{div}\,\bm \xi$, for the two--qubit Schmidt family $|\psi(\theta)\rangle=\cos\theta\,|00\rangle+\sin\theta\,|11\rangle$. The trace is positive and diverges at $\theta\to 0,\pi/2$ (almost product states) and at $\theta=\pi/4$ (maximally entangled state), indicating very strong extrinsic curvature of the constant–entanglement hypersurfaces in the Fubini-Study geometry near both separable and maximally entangled configurations. For intermediate $\theta$ the curvature is finite and relatively small, corresponding to a slower variation of the geometric entropy with entanglement.