Classicality of spin states

We extend the concept of classicality in quantum optics to spin states. We call a state “classical” if its density matrix can be decomposed as a weighted sum of angular momentum coherent states with positive weights. Classical spin states form a convex set $\mathcal{C}$, which we fully characterize for a spin $1∕2$ and a spin 1. For arbitrary spin, we provide “nonclassicality witnesses.” For bipartite systems, $\mathcal{C}$ forms a subset of all separable states. A state of two spins $1∕2$ belongs to $\mathcal{C}$ if and only if it is separable, whereas for a spin $1∕2$ coupled to a spin 1, there are separable states which do not belong to $\mathcal{C}$. We show that in general the question whether a state is in $\mathcal{C}$ can be answered by a linear programming algorithm.

Figure 1

(Color online) Example of a set of classical states $\mathcal{C}$ for a bipartite system of two spins $1∕2$ and 1 parametrized by two parameters, $\rho ={\rho }_0+{\kappa }_1{\widehat{\rho }}_1+{\kappa }_2{\widehat{\rho }}_2$ with some traceless ${\widehat{\rho }}_1,{\widehat{\rho }}_2$. Boundaries are shown of non-negativity of $\rho$ (bold black line), non-negativity of its partial transpose ${\rho }^{T_A}$ (dashed line), and of $P$ representability of $\rho ,{\rho }^{T_A}$ (inner red line).