Relevance of Bell's theorem as a signature of nonlocality: Case of classical angular momentum distributions

For a system composed of two particles, Bell’s theorem asserts that averages of physical quantities determined from local variables must conform to a family of inequalities. In this work we show that a classical model containing a local probabilistic interaction in the measurement process can lead to a violation of the Bell inequalities. We first introduce two-particle phase-space distributions in classical mechanics constructed to be the analogs of quantum-mechanical angular momentum eigenstates. These distributions are then employed in four schemes characterized by different types of detectors measuring the angular momenta. When the model includes an interaction between the detector and the measured particle leading to ensemble dependencies, the relevant Bell inequalities are violated if the total angular momentum is required to be conserved. The violation is explained by identifying assumptions made in the derivation of Bell’s theorem that are not fulfilled by the model, in particular noncommutativity of single-particle measurements. We discuss to what extent a violation of these assumptions is a faithful marker of nonlocality.

Figure 1

(Color online). Normalized angular distribution for a single particle in configuration space. (a) Classical distribution $\rho \left(\theta ,\varphi \right)$ of Eq. (2). (b) Corresponding quantum distribution [spherical harmonic ${\left|Y_{JM}\left(\theta ,\varphi \right)\right|}^2$ with $J/\eta =40$, $\eta =\hslash$, and $M/J=5/8$ as in (a)]. (c) Distribution of the angular momentum on the sphere for a distribution of the type $\rho \left(\theta ,\varphi \right)$, invariant around the $\mathit{z}$ axis with a fixed value of ${\mathit{J}}_{\mathit{z}}$.

Figure 2

(Color online). (a) Uniform distribution of ${\mathbf{J}}_1$ and ${\mathbf{J}}_2$ on the unit sphere; the angular momenta are correlated via the conservation law (5) and must thus point in opposite directions. In the first example (Sec. 3B), the detectors measure $J_{1a}$ and the correlated $J_{2b}$ as the angular momenta span the sphere. (b) Example 2 (Sec. 3C): Distribution of ${\mathbf{J}}_2$, when $D_{1a}=-1/2$ was obtained. A measurement of $D_{2b}$ will yield $\pm 1/2$ depending on the position of ${\mathbf{J}}_2$: if ${\mathbf{J}}_2$ lies within the light shaded region [intersection of the two positive hemispheres centered on $\mathit{a}$ and on $\mathit{b}$, denoted $\mathcal{D}\left(-1/2,1/2\right)$ in the text], $D_{2b}=1/2$ will be obtained, $-1/2$ when ${\mathbf{J}}_2$ belongs to the dark-shaded region $\left[\mathcal{D}\left(-1/2,-1/2\right)\right]$.

Figure 3

(Color online). (a) The ensemble ${\rho }_{1a+}$ for the single-particle model described in Sec. 4A. Any ${\mathbf{J}}_1$ in this ensemble has a positive projection $J_{1a}$; measuring $R_{1a}$ gives the outcome $+1/2$ with certainty, without changing the ensemble, since ${\rho }_{1a+}$ is symmetric relative to the $\mathit{a}$ axis and ${⟨J_{1a}⟩}_{{\rho }_{1a+}}=1/2$. (b) In the same situation $R_{1b}$ is measured. Now the symmetry axis of the ensemble ${\rho }_{1a+}$ does not coincide with the $\mathit{b}$ axis. Hence measuring $R_{1b}$ can yield either $+1/2$ or $-1/2$ with probabilities proportional to ${⟨H\left(\pm J_{1b}\right)J_{1b}⟩}_{{\rho }_{1a+}}$. The ensemble is modified during the measurement, undergoing a rotation toward the positive or negative $\mathit{b}$ axis as indicated by the arrows. (c) The two-particle distribution after $R_{1a}$ was measured and the outcome is known to have been $R_{1a}=-1/2$, in which case particle 1 is described by the ensemble ${\rho }_{1a-}$. Conservation of the angular momentum then requires that particle 2 be described by ${\rho }_{2a+}$, so that if $R_{2a}$ were measured, the outcome $+1/2$ would be obtained with certainty [Eq. (58)]. If instead $R_{2b}$ is measured, we have a single-particle problem for particle 2 identical to the one portrayed in Fig. 2b.