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arXiv:quant-ph/0703251 · Journal of Physics A Mathematical and Theoretical

Classical statistical distributions can violate Bell-type inequalities

A. Matzkin

Published Feb 12, 2008• 11 citations
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Abstract

We investigate two-particle phase-space distributions in classical mechanics constructed to be the analogs of quantum-mechanical angular-momentum eigenstates. We obtain the phase-space averages of specific observables related to the projection of the particles' angular momentum along axes with different orientations, and show that the ensuing correlation function violates Bell's inequality. The key to the violation resides in choosing observables impeding the realization of the joint measurements whose existence is required in the derivation of the inequalities. This situation can have statistical (detection related) or dynamical (interaction related) underpinnings, but non-locality does not play any role.

DOI
10.1088/1751-8113/41/8/085303

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We investigate two-particle phase-space distributions in classical mechanics constructed to be the analogs of quantum-mechanical angular-momentum eigenstates. We obtain the phase-space averages of specific observables related to the projection of the particles' angular momentum along axes with different orientations, and show that the ensuing correlation function violates Bell's inequality. The key to the violation resides in choosing observables impeding the realization of the joint measurements whose existence is required in the derivation of the inequalities. This situation can have statistical (detection related) or dynamical (interaction related) underpinnings, but non-locality does not play any role.
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.
We consider a system of two particles, each with large angular momentum jjj, in the singlet state. The probabilities of finding projections of the angular momenta on selected axes are determined. The generalized Bell inequalities involve these probabilities and we study them using statistical methods. We show that most of Bell's inequalities cannot be violated, or are violated only marginally, in the limit $j\ensuremath{\rightarrow}\ensuremath{\infty}$. The precision required to confirm a violation appears to be difficult to achieve. In practice, the quantum system, in spite of being entangled, becomes indistinguishable from its classical counterpart.
I derive the correlation function for a general theory of two-valued spin variables that satisfy the fundamental conservation law of angular momentum. The unique theory-independent correlation function is identical to the quantum mechanical correlation function. I prove that any theory of correlations of such discrete variables satisfying the fundamental conservation law of angular momentum violates the Bell’s inequalities. Taken together with the Bell’s theorem, this result has far reaching implications. No theory satisfying Einstein locality, reality in the EPR-Bell sense, and the validity of the conservation law can be constructed. Therefore, all local hidden variable theories are incompatible with fundamental symmetries and conservation laws. Bell’s inequalities can be obeyed only by violating a conservation law. The implications for experiments on Bell’s inequalities are obvious. The result provides new insight regarding entanglement, and its measures. PACS Numbers: 03.65.Ta, 03.65.Ud The main result of this paper is the proof that a general physical theory of correlations of two-valued spin projections violate the Bell’s inequalities if the theory satisfies the conservation law for angular momentum on the average. The proof is applicable to local hidden variable theories, as well as
Abstract. Bell’s theorem is a statement by which averages obtained from specific types of statistical distributions must conform to a family of inequalities. These models, in accordance with the EPR argument, provide for the simultaneous existence of quantum mechanically incompatible quantities. We first recall several contradictions arising between the assumption of a joint distribution for incompatible observables and the probability structure of quantum-mechanics, and conclude that Bell’s theorem is not expected to be relevant to quantum phenomena described by non-commuting observables, irrespective of the issue of locality. Then, we try to disentangle the locality issue from the existence of joint distributions by introducing two models accounting for the EPR correlations but denying the existence of joint distributions. We will see that these models do not need to resort explicitly to non-locality: the first model relies on conservation laws for ensembles, and the second model on an equivalence class by which different configurations lead to the same physical predictions. Keywords: Quantum mechanics, Non-locality, Bell inequalities
Nonlocal effects associated with quantum-mechanical states have been theoretically interesting because they emphasize the differences between classical and quantum mechanics. We describe a system in which a single input photon causes a nonlocal influence on two spatially separated homodyne detectors, as evidenced by a violation of Bell's inequalities. Unlike previous configurations for violating Bell's inequalities, the correlation is not imposed on a pair of particles that spatially separate before being detected. Rather, the correlation arises from the interaction of the local oscillators at each homodyne detector with the spatially extended field of the input photon. The above effect is only one of a class of situations in which nonlocal correlations can be imposed on separated homodyne detectors by quantum mechanical fields. By deriving the conditions for violating Bell's inequality with phase-sensitive detectors, new tests can be developed that relate to squeezed light and other nonclassical fields.
Bell's theorem is a statement by which averages obtained from specific types of statistical distributions must conform to a family of inequalities. These models, in accordance with the EPR argument, provide for the simultaneous existence of quantum mechanically incompatible quantities. We first recall several contradictions arising between the assumption of a joint distribution for incompatible observables and the probability structure of quantum-mechanics, and conclude that Bell's theorem is not expected to be relevant to quantum phenomena described by non-commuting observables, irrespective of the issue of locality. Then, we try to disentangle the locality issue from the existence of joint distributions by introducing two models accounting for the EPR correlations but denying the existence of joint distributions. We will see that these models do not need to resort explicitly to non-locality: the first model relies on conservation laws for ensembles, and the second model on an equivalence class by which different configurations lead to the same physical predictions.