Nonclassicality in the statistics of noncommuting observables: Nonclassical states are more compatible than classical states

We study nonclassicality in the product of the probabilities of noncommuting observables. We show that within the quantum theory, nonclassical states can provide larger probability product than classical states, so that nonclassical states approach the nonfluctuating states of the classical theory more closely than classical states. This is particularized to relevant complementary observables such as conjugate quadratures, phase and number, quadrature and number, and orthogonal angular momentum components.

Figure 1

Diagram sketching in the upper panel the standard intuition regarding the closeness of the classical states within the quantum theory to the states without fluctuations of the classical theory. The results of this work are illustrated in the lower part.

Figure 2

Plot of the numerical evaluation of the exact $p(n,\varphi |\alpha )$ for $\arg \alpha =\varphi$ (solid line) and its approximation in Eq. (25) (dashed line) for $n=0,5,10$ as functions of $\left|\alpha \right|{}^2$.

Figure 3

Plot of $p(n=0,\varphi =\arg \varepsilon |\varepsilon )$ (solid line) and the classical upperbound (26) $p(n=0,\varphi |{\Phi }_{n=0,\varphi })$ (dashed line) as functions of $\left|\varepsilon \right|$ showing that there is nonclassical behavior for $\left|\varepsilon \right|>0.47$.

Figure 4

Plot of the individual number $p(n=0|\varepsilon )$ (solid line) and phase $p(\varphi =\arg \varepsilon |\varepsilon )$ (dashed line) probabilities for the phase coherent state $|\varepsilon \rangle$ as functions of $\left|\varepsilon \right|$.

Figure 5

Plot of the $Q(\alpha )$ function with $\alpha =x/2+\mathit{iy}/2$ for a phase coherent state $|\varepsilon \rangle$ with $\varepsilon =0.2$ (left) and $\varepsilon =0.9$ (right) showing that it becomes more elongated as $\left|\varepsilon \right|$ increases.

Figure 6

Plot of $p(n=0,\varphi =0|\xi )$ (solid line) and the classical upper bound $p(n=0,\varphi |{\Phi }_{n=0,\varphi })$ (dashed line) as a function of $R$ for a squeezed coherent state with fixed mean number of photons $\langle a^{†}a\rangle =10$.

Figure 7

Plot of the individual number $p(n=0|\xi )$ (solid line) and phase $p(\varphi =\arg \xi |\xi )$ (dashed line) probabilities for the quadrature squeezed state $|\xi \rangle$ with fixed mean number of photons $\langle a^{†}a\rangle =10$ as functions of $R$.

Figure 8

Plot of the $Q$ function of the squeezed coherent states $|\xi \rangle$ with fixed mean number of photons $\langle a^{†}a\rangle$ and coherent amplitudes $R=0$ (left) and $R=2.8$ (right).

Figure 9

Plot of the product of probabilities $p(j,j|\psi )$ for the state in Eq. (47) (solid line), and the classical upperbound $p(j,j|{\Phi }_{j,j})$ in Eq. (46) (dashed line), as functions of $j$ showing nonclassical behavior for $j>1/2$.