Anticoherence of spin states with point-group symmetries

We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana representation of spin states in terms of points on the unit sphere, we analyze the consequences of a point-group symmetry in their arrangement on the quantum properties of the corresponding state. We focus on the identification of anticoherent states (for which all reduced density matrices in the symmetric subspace are maximally mixed) associated with point-group-symmetric sets of points. We provide three different characterizations of anticoherence and establish a link between point symmetries, anticoherence, and classes of states equivalent through stochastic local operations with classical communication. We then investigate in detail the case of small numbers of qubits and construct infinite families of anticoherent states with point-group symmetry of their Majorana points, showing that anticoherent states do exist to arbitrary order.

Figure 1

Husimi function of the 12-qubit 5-anticoherent state $|{\psi }_S\rangle =\frac{\sqrt{7}}{5}|D_{12}^{\left(1\right)}\rangle +\frac{\sqrt{11}}{5}|D_{12}^{\left(6\right)}\rangle -\frac{\sqrt{7}}{5}|D_{12}^{\left(11\right)}\rangle$. The Majorana points, depicted by white points on the sphere, display an icosahedral symmetry. The color code is chosen so as to highlight the contour lines of the Husimi function.

Figure 2

Majorana representation of five-qubit 1-anticoherent states displaying $C_n$ symmetry. The small and large points on the spheres correspond to nondegenerate and twice-degenerate Majorana points, respectively. The corresponding states are those given in the main text (see Sec. 4b2).

Figure 3

Husimi function of $C_n$-symmetric states corresponding to the smallest possible number $N_t$ of qubits for order of anticoherence $t=2,...,7$, with $n>t$. From left to right: 2-anticoherent state of 4 qubits (tetrahedron, $C_3$ symmetry), 3-anticoherent state of 6 qubits (octahedron, $C_4$ symmetry), 4-anticoherent state of 12 qubits $(C_5$ symmetry), 5-anticoherent state of 24 qubits $(C_6$ symmetry), 6-anticoherent state of 42 qubits $(C_7$ symmetry), 7-anticoherent state of 75 qubits $(C_8$ symmetry).

Figure 4

Each bar corresponds to a couple $(t,N)$ for which a $C_n$-symmetric $N$-qubit $t$-anticoherent state with $t<n$ exists. In particular, the minimum number of qubits $N_t$ for which such a state exists is given by the position of the first bar on each line. This value $N_t$ is well fitted by $N_t=t(at+b)$, with $a=1.52$ and $b=-3.93$ (dashed line). Color (grayness) represents the number $r=(N-n_S-n_N)/n$ of free Dicke coefficients (all others being zero) on which the linear optimisation is realized.

Figure 5

Husimi function of the 42-qubit 7-anticoherent state $|{\psi }_S\rangle =\left(\sqrt{7062}\right|D_{42}^{\left(0\right)}\rangle +\sqrt{29315}|D_{42}^{\left(7\right)}\rangle +3\sqrt{451}|D_{42}^{\left(14\right)}\rangle +\sqrt{36777}|D_{42}^{\left(21\right)}\rangle -3\sqrt{451}|D_{42}^{\left(28\right)}\rangle +\sqrt{29315}|D_{42}^{\left(35\right)}\rangle -\sqrt{7062}|D_{42}^{\left(42\right)}\rangle )/343$. This state displays $D_{7d}$ symmetry; hence, $n=t=7$.