Multiqubit symmetric states with maximally mixed one-qubit reductions

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.

Figure 1

Majorana representation of the MES states $|{\psi }_{\mu }\rangle$ [Eq. (25)] belonging to the 4-qubit SLOCC classes ${\mathcal{C}}_{\mu }^{1,1,1,1}$ [$\mu \in \mathit{S},(\theta ,\phi )\in {\mathit{S}}^{\prime }$].

Figure 2

Density plot of the Husimi function (17) associated with the tetrahedron state $|T_4\rangle \equiv |{\psi }_{\mu =i\sqrt{2}}\rangle$ (black dots are the Majorana points of the state, two being only visible in the picture).

Figure 3

Density plot of the geometric measure of entanglement $E_G$ of the 4-qubit MES states $|{\psi }_{\mu }\rangle$ [Eq. (25)] as a function of the real and imaginary parts of $\mu \in \mathit{S}$ [Eq. (26)], the only region where distinct $\mu$ define SLOCC-inequivalent ${\mathcal{D}}_{1,1,1,1}$ states (see text). The geometric entanglement is constant along the dashed curves. These curves cross the region boundaries at right angles. Particular values of $\mu$ are highlighted: $|{\psi }_{\mu =0}\rangle =|{\mathrm{GHZ}}_4\rangle ,|{\psi }_{\mu =i\sqrt{2}}\rangle \equiv |T_4\rangle$ (tetrahedron state). The dash-dotted arc of circle separates two regions I and II where the density plots of $E_G$ are just the image of each other through the conjugated Moebius transformation $\mu \to {[2(\sqrt{6}-\mu )/(\sqrt{6}\mu +2)]}^{*}$ (see text).

Figure 4

Barycentric measure of entanglement $E_B$ of the MES states $|P_N\rangle$ (blue dots) and the SLOCC equivalent non-MES states $|P_{N,\alpha }\rangle \left(\alpha =-\sqrt[8]{27/25}\right)$ (orange triangles) as a function of the number of qubits $N$.