An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglementof two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.
An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglementof two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.
One of the biggest problems in the theory of quantum information is the quantification of amount of entanglement in an arbitrary multipartite mixed state. Different axiomatic and operational measures were proposed so far. In this work we will establish a connection between geometric measure of entanglement proposed in [Phys. Rev. A 68, 042307 (2003)] and measures based on fidelity. One result will be, that geometric and revised geometric measure of entanglement proposed in [J. Phys. A: Math. Theor. 40, 3507 (2007)] are equal. Further a useful expression for fidelity is derived.
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
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.
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.
Several inequivalent definitions of the geometric measure of entanglement (GM) have been introduced and studied in the past. Here we review several known and new definitions, with the qualifying criterion being that for pure states the measure is a linear or logarithmic function of the maximal fidelity with product states. The entanglement axioms and properties of the measures are studied, and qualitative and quantitative comparisons are made between all definitions. Streltsov et al. [New J. Phys. 12, 123004 (2010)] proved the equivalence of two linear definitions of GM, whereas we show that the corresponding logarithmic definitions are distinct. Certain classes of states such as ``maximally correlated states'' and isotropic states are particularly valuable for this analysis. A little-known GM definition is found to be the first one to be both normalized and weakly monotonous, thus being a prime candidate for future studies of multipartite entanglement. We also find that a large class of graph states, which includes all cluster states, have a ``universal'' closest separable state that minimizes the quantum relative entropy, the Bures distance, and the trace distance.
ABSTRACT We investigate entanglement in two‐qubit systems using a geometric representation based on the minimum of essential parameters. The latter is achieved by requiring subsystems with the same entropy, regardless of whether the state of the entire system is pure or mixed. The geometric framework is provided by a convex set that forms a right‐triangle, whose points are linked to the absolute value of just two of the coherences of the system under study. As a result, we find that optimized states of two qubits host pairs of identical populations while reducing the number of coherences involved, so they are X‐shaped. A geometric ‐measure of entanglement is introduced as the distance between the points in that represent entangled states and the closest point that defines separable states. It is shown that reproduces the results of the Hill‐Wootters concurrence , so that can be interpreted as a distance‐like entanglement measure. However, unlike , the measure also distinguishes the rank of states. The universality of the two‐qubit X‐states ensures the utility of our geometric model for studying entanglement of two‐qubit states in any configuration. To show the applicability of our approach far beyond time‐independent cases, we construct a time‐dependent two‐qubit state, traced out over the complementary components of a pure tetra‐partite system, and find that its one‐qubit states share the same entropy. The entanglement measure results bounded from above by the envelope of the minima of such entropy.