Absolutely classical spin states

We introduce the concept of “absolutely classical” spin states, in analogy to absolutely separable states of bipartite quantum systems. Absolutely classical states are states that remain classical (i.e., a convex sum of projectors on coherent states of a spin $j$) under any unitary transformation applied to them. We investigate the maximal size of the ball of absolutely classical states centered on the maximally mixed state and derive a lower bound for its radius as a function of the total spin quantum number. We also obtain a numerical estimate of this maximal radius and compare it to the case of absolutely separable states.

Figure 1

Maximal radius $r_{max}\left(j\right)$ of a ball of classical states centered at the fully mixed state as a function of $j$. Blue dots: the value of the lower bound $\equiv {\widehat{r}}_{\max }\left(j\right)$; Eq. (18). Black crosses: smallest numerically found distance from the maximally mixed state to a nonclassical state. Red line: maximal ball size $1/\left(2^j\sqrt{4^j-1}\right)$ for arbitrary (not necessarily symmetric) separable states [20]. This function gives an excellent approximation of the numerically found maximal ball size ${\tilde{r}}_{\max }\left(j\right)$ of classical spin-$j$ states, but slightly overestimates it for small $j$.