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.
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.
Entanglement Distance (ED) was originally proposed as a geometric measure of entanglement derived from the Fubini-Study metric on the projective Hilbert space. Independently, the Meyer-Wallach and Scott measures quantify multipartite entanglement via linear entropy. In this work, we demonstrate that these two seemingly distinct frameworks are mathematically identical for pure states of arbitrary finite dimensions. We prove that ED arises naturally as the trace of the Fubini-Study metric tensor over the local subalgebra of observables. Crucially, this geometric unification yields a direct operational interpretation: the global entanglement of a pure state is exactly proportional to the total Quantum Fisher Information (QFI) available for local unitary estimation. This bridges abstract information geometry with quantum metrology, demonstrating that ED dynamically quantifies resourcefulness for distributed quantum sensing, identifying Heisenberg-limited sensitivity in regimes where standard variance-based witnesses fail.
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.
We show that the manifold of quantum states is endowed with a rich and nontrivial geometric structure. We derive the Fubini–Study metric of the projective Hilbert space of a multi-qubit quantum system, endowing it with a Riemannian metric structure, and investigate its deep link with the entanglement of the states of this space. As a measure, we adopt the entanglement distance E preliminary proposed in Phys. Rev. A 101, 042129 (2020). Our analysis shows that entanglement has a geometric interpretation: E(∣ψ〉) is the minimum value of the sum of the squared distances between ∣ψ〉 and its conjugate states, namely the states νμ · σμ∣ψ〉, where νμ are unit vectors and μ runs on the number of parties. Within the proposed geometric approach, we derive a general method to determine when two states are not the same state up to the action of local unitary operators. Furthermore, we prove that the entanglement distance, along with its convex roof expansion to mixed states, fulfils the three conditions required for an entanglement measure, that is: i) E(∣ψ〉) = 0 iff ∣ψ〉 is fully separable; ii) E is invariant under local unitary transformations; iii) E does not increase under local operation and classical communications. Two different proofs are provided for this latter property. We also show that in the case of two qubits pure states, the entanglement distance for a state ∣ψ〉 coincides with two times the square of the concurrence of this state. We propose a generalization of the entanglement distance to continuous variable systems. Finally, we apply the proposed geometric approach to the study of the entanglement magnitude and the equivalence classes properties, of three families of states linked to the Greenberger–Horne–Zeilinger states, the Briegel Raussendorf states and the W states. As an example of application for the case of a system with continuous variables, we have considered a system of two coupled Glauber coherent states.
A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.
Abstract Computationally feasible multipartite entanglement measures are essential for advancing our understanding of complex quantum systems. Entanglement distance (ED), introduced by Cocchiarella et al (2020 Phys. Rev. A 101 042129), based on the Fubini–Study metric, offers several advantages over existing methods, including ease of computation, a profound geometrical interpretation, and applicability to multipartite entanglement. Although ED has been successfully applied to systems of qudits, an explicit formulation for continuous quantum variables, particularly for pure Gaussian states, remains unexplored. In this work, we address this limitation by deriving the analytical expression for the Gaussian entanglement measure (GEM), a multipartite entanglement monotone for multimode pure Gaussian states based on the purity of fragments of the whole system, through a generalization of ED to group-theoretic coherent states. We show the efficacy of GEM across various scenarios, including the analysis of a two-mode Gaussian state under beam-splitter and squeezing transformations, and exploring graph states involving three and four modes. Notably, comparing GEM values for states with different graph topologies reveals insights into the connectivity of the underlying graphs. Additionally, we illustrate how GEM provides insights into free bosonic field theory on R t × S 1 beyond standard bipartite entanglement entropy, paving the way for quantum information-theoretic tools to probe the topological properties of quantum field theories.
We develop a comprehensive axiomatic framework for quantum-inspired distance metrics on projective Hilbert spaces, providing a unified foundation that organizes and generalizes existing measures in quantum information theory. Starting from five fundamental axioms, projective invariance, unitary covariance, superposition sensitivity, entanglement awareness, and measurement contextuality, we show that any admissible distance depends solely on state overlap and establish the uniqueness of the Fubini-Study metric as the canonical geodesic distance. Our framework further yields a hierarchy of comparison results relating the Fubini-Study metric, Bures distance, Euclidean distance, measurement-based pseudometrics, and entanglement-sensitive distances. Key contributions include an entanglement-geometry complementarity principle, high-dimensional concentration bounds, and operational interpretations connecting distances to state discrimination and quantum metrology. This work places the geometry of quantum state spaces on a rigorous axiomatic footing, bridging abstract metric theory, information geometry, and operational quantum principles.