It is widely recognized that concurrence can be regarded as a measure of two-qubit entanglement. This paper studies the geometry of concurrence for a two-qubit system to show that the concurrence serves as a coordinate of the factor space , where is the space of normalized two-qubit states, and where G = U (1) × SU (2) × SU (2). Any monotonically increasing function of the concurrence can serve as a measure for entanglement. From the viewpoint of Riemannian geometry, a state with concurrence r is shown to be distant from the separable states by , where r ranges over 0 ⩽ r ⩽ 1. In addition, measures of entanglement for n -qubits are discussed on the basis of a bipartite decomposition with ℓ + m = n . They are invariant under the local unitary transformation group U (2 ℓ ) × U (2 m ).
It is widely recognized that concurrence can be regarded as a measure of two-qubit entanglement. This paper studies the geometry of concurrence for a two-qubit system to show that the concurrence serves as a coordinate of the factor space , where is the space of normalized two-qubit states, and where G = U (1) × SU (2) × SU (2). Any monotonically increasing function of the concurrence can serve as a measure for entanglement. From the viewpoint of Riemannian geometry, a state with concurrence r is shown to be distant from the separable states by , where r ranges over 0 ⩽ r ⩽ 1. In addition, measures of entanglement for n -qubits are discussed on the basis of a bipartite decomposition with ℓ + m = n . They are invariant under the local unitary transformation group U (2 ℓ ) × U (2 m ).
This paper, a continuation of a previous one ( Iwai 2007 J. Phys. A: Math. Theor. 40 1361 ), studies the geometry of multi-qubit entanglement with respect to bipartite partitions. An n -qubit system ( C 2 ) ⊗ n is isomorphic with , the linear space of 2 ℓ × 2 m complex matrices, where ℓ + m = n . According to the isomorphism, the local transformation group U (2 ℓ ) × U (2 m ) acts on the space of normalized states in . Let M and G denote the space of normalized states and the local transformation group acting on M , respectively. According to the orbit types, M is stratified into strata, among which a principal stratum and the sets of separable states and maximally entangled states will be identified. For , a function F ( C ) = det( I − CC *) proves to serve as a measure of entanglement, where I denotes the 2 ℓ × 2 ℓ identity matrix. The F ( C ) attains the minimal and the maximal values, respectively, on the sets of separable states and maximally entangled states, and further takes no extremal values on the principal stratum. The F ( C ) projects to a function on the factor space G \ M . A naturally defined metric on M also projects to that on G \ M , which serves to measure the distance between the separable states, F −1 (0), and the states, F −1 ( k ), of prescribed value k of measure. To be precise, one has to restrict M to the principal stratum, when projecting the metric. Three- and four-qubit systems will be studied intensively, and then multi-qubit systems discussed.
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.
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.
The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.
Concurrence is a crucial entanglement measure in quantum theory used to describe the degree of entanglement between two or more qubits. Local unitary (LU) invariants can be employed to describe the relevant properties of quantum states. Compared to quantum state tomography, observing LU invariants can save substantial physical resources and reduce errors associated with tomography. In this paper, we use LU invariants as explanatory variables and employ methods such as multiple regression, tree models, and BP neural network models to fit the concurrence of 2-qubit quantum states. For pure states and Werner states, by analyzing the correlation between data, a functional formula for concurrence in terms of LU invariants is obtained. Additionally, for any two-qubit quantum states, the prediction accuracy for concurrence reaches 98.5%.
The behavior of quantum states under local unitary transformations (LUTs) and stochastic local quantum operations and classical communication (SLOCC) has proven central to the understanding of entanglement in multipartite quantum systems. In particular, invariants under these operations have provided insight into quantum entanglement in multiple‐qubit states. Relationships between entanglement, mixedness and spin symmetry in multiple‐qubit quantum states can be found by exploiting these properties. For example, concurrence and n‐tangle are naturally expressed in terms of spin‐flip transformations. In the case of specialized entanglement measures and/or special families of states, complementarity relations involving these lengths and state transformations are derivable. Here, the role of geometry in such investigations is explored. Minkowskian geometry is seen to provide elegant representations of entanglement in multiple‐qubit systems, particularly the language of twistors.