Geometric multiaxial representation of $N$-qubit mixed symmetric separable states

The study of $N$-qubit mixed symmetric separable states is a longstanding challenging problem as no unique separability criterion exists. In this regard, we take up the $N$-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer an elegant mathematical analysis since the dimension of the Hilbert space is reduced from $2^N$ to $N+1$. Since there exists a one-to-one correspondence between the spin-$j$ system and an $N$-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of the spin-density matrix. Further, we use a geometric multiaxial representation (MAR) of the density matrix to characterize the mixed symmetric separable states. Since the separability problem is NP-hard, we choose to study it in the continuum limit where mixed symmetric separable states are characterized by the $P$-distribution function $\lambda (\theta ,\varphi )$. We show that the $N$-qubit mixed symmetric separable states can be visualized as a uniaxial system if the distribution function is independent of $\theta$ and $\varphi$. We further choose a distribution function to be the most general positive function on a sphere and observe that the statistical tensor parameters characterizing the $N$-qubit symmetric system are the expansion coefficients of the distribution function. As an example for the discrete case, we investigate the MAR of a uniformly weighted two-qubit mixed symmetric separable state. We also observe that there exists a correspondence between the separability and classicality of states.

Figure 1

Multiaxial representation of mixed symmetric separable states showing the axes characterized by $t_0^k$ for $k=1,2$.