Quantumness of spin-1 states

We investigate quantumness of spin-1 states, defined as the Hilbert-Schmidt distance to the convex hull of spin coherent states. We derive its analytic expression in the case of pure states as a function of the smallest eigenvalue of the Bloch matrix and give explicitly the closest classical state for an arbitrary pure state. Numerical evidence is given that the exact formula for pure states provides an upper bound on the quantumness of mixed states. Due to the connection between quantumness and entanglement we obtain new insights into the geometry of symmetric entangled states.

Figure 1

The Majorana representation of the pure state $|{\psi }_{\gamma }\rangle$ given through Eq. (9), shown for $\gamma =\frac{\pi }{4}$, which corresponds to $\lambda =-\frac{1}{3}$.

Figure 2

Quantumness of randomly generated mixed states, as a function of the smallest eigenvalue of their Bloch matrix. There are 50 000 points, each one corresponding to a random state. Red line corresponds to the pure state analytic result $f\left(\lambda \right)$ given by Eq. (38). Dashed line indicates the lower bound (45). Function $f\left(\lambda \right)$ appears to be an upper bound on the quantumness of mixed states (see inset).

Figure 3

Difference between the quantumness and the hypothetical upper bound $f\left(\lambda \right)$ as a function of the smallest eigenvalue of the Bloch matrix (same data as in Fig. 2). The difference between the upper bound and the quantumness is of the order of ${10}^{-3}$. The numerical error is of order ${10}^{-6}$, and our numerical procedure can only overestimate quantumness so that the points could only be lower than they appear here by that amount. The dashed line corresponds to the lower bound (45).

Figure 4

Visualization of the allowed parameter range of the variables $u$ and $v$. The upper area corresponds to $u\ge 1-{\lambda }^2$, while the lower corresponds to $u\le 1-{\lambda }^2$. We call the function $F(u,v,g)$ restricted to the upper (lower) area $D\left(E\right)$.