Angular-momentum nonclassicality by breaking classical bounds on statistics

We derive simple practical procedures revealing the quantum behavior of angular momentum variables by the violation of classical upper bounds on the statistics. Data analysis is minimum and definite conclusions are obtained without evaluation of moments, or any other more sophisticated procedures. These nonclassical tests are very general and independent of other typical quantum signatures of nonclassical behavior such as sub-Poissonian statistics, squeezing, or oscillatory statistics, being insensitive to the nonclassical behavior displayed by other variables.

Figure 1

Plot of $p_{j,m}$ (diamonds joined by a solid line) and ${\tilde{\mathcal{P}}}_{j,\rho }$ (dashed line) for a phase-averaged SU(2) coherent state with $j=10$ showing that the classical bound is clearly infringed by the statistics of the outcomes $m=0,\pm 1$.

Figure 2

Plot of ${\mathcal{P}}_{j,m}$ (diamonds joined by a solid line) and ${\mathcal{P}}_{j,m}^{\prime }$ (stars joined by a dotted line) for $j=10$ as functions of $m$ to illustrate their difference.

Figure 3

Plot of the $j_1$ statistics $p_{j,m}$ (diamonds joined by solid line) for the Schrödinger cat state (3.21) and the SU(2) bound for classical states ${\mathcal{P}}_{j,m}$ (stars joined by a dotted line) in Eq. (3.5) for $j=10$ showing that for $m=0,\pm 2$ there is a clear violation of the classical upper bound.

Figure 4

Plot of the statistics of $j_{\perp ,1}$ (diamonds joined by solid line) for the state (3.24) for $j=10$, $\eta =0.5$ along with the SU(2) classical state upper bound (3.5) (stars joined by dotted line) showing the violation of the classical bounds for small $\left|m\right|$.

Figure 5

Plot of the probability $p_{j,m=0}$ (solid line) of the eigenvalue $m=0$ of $j_{\perp ,1}$ for the state (3.24) with $j=10$ as a function of $\eta$ along with the SU(2) classical upper bound (3.6) ${\mathcal{P}}_{j,m=0}$ (dashed line) showing nonclassical behavior for all $\eta <1$.