Abstract We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. https://doi.org/10.1103/PhysRevA.95.032308 ), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces.
We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity
G(ρ)=[1−L(ρ)]/L(ρ) from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree
Q(ρ)=[(1−p)/p]/G(ρ) by comparing it with the robustness relative to the maximally mixed state.
We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound
(1−p)LB/[p(1−LB)]≤p0≤Q(ρ) and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely
[1−LR]/LR≤[1−LB]/LB.
For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound
(1−p)/(2p)≤p0≤Q(ρ). These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.
The basic question that is addressed in this paper is finding the closest separable state for a given entangled state, measured with the Hilbert-Schmidt distance. While this problem is in general very difficult, we show that the following strongly related problem can be solved: find the Hilbert-Schmidt distance of an entangled state to the set of all partially transposed states. We prove that this latter distance can be expressed as a function of the negative eigenvalues of the partial transpose of the entangled state, and show how it is related to the distance of a state to the set of positive, partially transposed (PPT) states. We illustrate this by calculating the closest biseparable state to the W state and give a simple and very general proof for the fact that the set of W-type states is not of measure zero. Next we show that all surfaces with states whose partial transposes have constant minimal negative eigenvalue are similar to the boundary of PPT states. We illustrate this with some examples on bipartite qubit states, where contours of constant negativity are plotted on two-dimensional intersections of the complete state space.
Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general $K\ifmmode\times\else\texttimes\fi{}M$ problem and characterize the set of effectively different states (which cannot be related by local transformations). Thus, we generalize earlier results obtained for the simplest $2\ifmmode\times\else\texttimes\fi{}2$ system, which lead to a stratification of the six-dimensional set of
N=4 pure states. We define the concept of absolutely separable states, for which all globally equivalent states are separable.
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.
The geometric measure of quantum entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to the geometric measure by means of a system of equations. The entanglement eigenvalue for a mixed state is introduced. And we show that, for a given mixed state, its nearest disentangled mixed state is associated with its entanglement eigenvalue. Two numerical examples are used to demonstrate the effectiveness of the proposed method.