We provide methods for computing the geometric measure of entanglement for two families of pure states with both experimental and theoretical interests: symmetric multiqubit states with non-negative amplitudes in the Dicke basis and symmetric three-qubit states. In addition, we study the geometric measure of pure three-qubit states systematically in virtue of a canonical form of their two-qubit reduced states and derive analytical formulas for a three-parameter family of three-qubit states. Based on this result, we further show that the W state is the maximally entangled three-qubit state with respect to the geometric measure.
The geometric measure of entanglement is the distance or angle between an entangled target state and the nearest unentangled state. Often one considers the geometric measure of entanglement for highly symmetric entangled states because it simplifies the calculations and allows for analytic solutions. Although some symmetry is required in order to deal with large numbers of qubits, we are able to loosen significantly the restrictions on the highly symmetric states considered previously, and consider several generalizations of the coefficients of both target and unentangled states. This allows us to compute the geometric entanglement measure for larger and more relevant classes of states.
We present a comprehensive study of maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. A general criterion is provided to easily identify whether given symmetric states are maximally entangled in that respect or not. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin of the system vanishes, as well as in corollary the dipole moment of the Husimi function. We establish the link between this kind of maximal entanglement, the anticoherence properties of spin states, and the degree of polarization of light fields. We analyze the relationship between the MES states and the classes of states equivalent through stochastic local operations with classical communication (SLOCC). We provide a nonexistence criterion of MES states within SLOCC classes of qubit states and show in particular that the symmetric Dicke state SLOCC classes never contain such MES states, with the only exception of the balanced Dicke state class for even numbers of qubits. The 4-qubit system is analyzed exhaustively and all MES states of this system are identified and characterized. Finally the entanglement content of MES states is analyzed with respect to the geometric and barycentric measures of entanglement, as well as to the generalized
N-tangle. We show that the geometric entanglement of MES states is ensured to be larger than or equal to
1/2, but also that MES states are not in general the symmetric states that maximize the investigated entanglement measures.
We study the geometric measure of entanglement (GM) of pure symmetric states related to rank 1 positive-operator-valued measures (POVMs) and establish a general connection with quantum state estimation theory, especially the maximum likelihood principle. Based on this connection, we provide a method for computing the GM of these states and demonstrate its additivity property under certain conditions. In particular, we prove the additivity of the GM of pure symmetric multiqubit states whose Majorana points under Majorana representation are distributed within a half sphere, including all pure symmetric three-qubit states. We then introduce a family of symmetric states that are generated from mutually unbiased bases and derive an analytical formula for their GM. These states include Dicke states as special cases, which have already been realized in experiments. We also derive the GM of symmetric states generated from symmetric informationally complete POVMs (SIC POVMs) and use it to characterize all inequivalent SIC POVMs in three-dimensional Hilbert space that are covariant with respect to the Heisenberg-Weyl group. Finally, we describe an experimental scheme for creating the symmetric multiqubit states studied in this article and a possible scheme for measuring the permanence of the related Gram matrix.
We introduce a class of generalized geometric measures of entanglement. For pure quantum states of
N elementary subsystems, they are defined as the distances from the sets of
K-separable states
(K=2,…,N). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows discrimination among the different contributions. The extended measures are applied to the study of entanglement in different classes of
N-qubit pure states. These classes include
W and Greenberger-Horne-Zeilinger (GHZ) states, and their symmetric superpositions; symmetric multimagnon states; cluster states; and, finally, asymmetric generalized
W-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between GHZ and
W states, clarifying the original result of Wei and Goldbart that
W states possess a larger global entanglement than GHZ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.
A method is developed to derive algebraic equations for the geometric measure of entanglement of three-qubit pure states. The equations are derived explicitly and solved in the cases of most interest. These equations allow one to derive analytic expressions of the geometric entanglement measure in a wide range of three-qubit systems, including the general class of W states and states which are symmetric under the permutation of two qubits. The nearest separable states are not necessarily unique, and highly entangled states are surrounded by a one-parametric set of equally distant separable states. A possibility for physical applications of the various three-qubit states to quantum teleportation and superdense coding is suggested from the aspect of entanglement.