Anticoherence measures for pure spin states

The set of pure spin states with vanishing spin expectation value can be regarded as the set of the less coherent pure spin states. This set can be divided into a finite number of nested subsets on the basis of higher order moments of the spin operators. This subdivision relies on the notion of anticoherent spin state to order $t$: A spin state is said to be anticoherent to order $t$ if the moment of order $k$ of the spin components along any directions are equal for $k=1,2,...,t$. Most spin states are neither coherent nor anticoherent, but can be arbitrary close to one or the other. In order to quantify the degree of anticoherence of pure spin states, we introduce the notion of anticoherence measures. By relying on the mapping between spin-$j$ states and symmetric states of $2j$ spin $1/2$ (Majorana representation), we present a systematic way of constructing anticoherence measures to any order. We briefly discuss their connection with measures of quantum coherence. Finally, we illustrate our measures on various spin states and use them to investigate the problem of the existence of anticoherent spin states with degenerated Majorana points.

Figure 1

Measures of 1 anticoherence (49) for the spin-1 state (48) as a function of $\theta$.

Figure 2

Density plot of the Bures measure of 2 anticoherence [Eq. (44)] of the 1-anticoherent spin-2 states (5) as a function of the real and imaginary parts of $\mu \in \mathit{D}$ [see Eq. (50)], the only region where distinct $\mu$ define states of the form (5) that are not connected by a rotation. This region includes the thick black borders. The measure ${\mathcal{A}}_2^{\mathrm{Bures}}$ is constant along the dashed curves. The minimum value, reached for $\mu =0$, is equal to $(1+\sqrt{2}-\sqrt{3})/2$. Particular values of $\mu$ are highlighted: $|{\psi }_2^{\mathrm{cat}}\rangle$ for $\mu =0$ [Eq. (1)], $|{\psi }_2^{\mathrm{tet}}\rangle$ for $\mu =i\sqrt{2}$ [Eq. (2)], and a state which is connected by rotation to the Dicke state $|2,0\rangle$ for $\mu =\sqrt{2/3}$.

Figure 3

Purity-based measure of 1 anticoherence (24) of the state (57) as a function of $\epsilon$ for different spin quantum numbers.

Figure 4

Purity-based (top) and Bures (bottom) measure of $t$-anticoherence of the state (57) for $j=10$ and $t=1,2,3,18,19$ as a function of $\epsilon$.

Figure 5

Measures of anticoherence as a function of the order $t$ for the spin-50 state $|{\psi }_{50}^{\mathrm{Coul}}\rangle$ whose Majorana points form an arrangement on the Bloch sphere identical to the one of point charges minimizing the Coulomb potential energy. Blue dots, ${\mathcal{A}}_t^R$; purple rhombus, ${\mathcal{A}}_t^{\mathrm{HS}}$; green squares, ${\mathcal{A}}_t^{\mathrm{tr}}$; and orange triangles, ${\mathcal{A}}_t^{\mathrm{Bures}}$.

Figure 6

Largest degeneracy degree $g_{\max }$ of at least one of the Majorana points for which $t$-anticoherent states with a spin quantum number $j$ are found numerically by optimization of the purity based measure of anticoherence ${\mathcal{A}}_t^R$. When no states are found, we set $g_{\max }=0$. In particular, the minimal spin quantum number $j_{\min }$ for which $t$-anticoherent states are found can be read from this figure: $j_{\min }=1$ for $t=1, j_{\min }=2$ for $t=2, j_{\min }=3$ for $t=3, j_{\min }=6$ for $t=4$, and $j_{\min }=6$ for $t=5$.

Figure 7

Majorana representation of states with minimal spin quantum number $j_{\min }$ for a given order $t$ of anticoherence and a given degeneracy degree $g$ of one of the Majorana points (the point located at the south pole). From top left to bottom right: (59) with $j=9/2$, (60) with $j=19/2$, (61), (A1), (A2), (62) with $\theta =0$, and (A3), (62) with $\theta =\pi /2$. The small dots represent nondegenerated Majorana points, while the large dots represent Majorana points with $g$-fold degeneracy as specified in the labels (we use the same color code as in Fig. 6).