[Paper Review] Quantum mechanical probabilities and general probabilistic constraints for Einstein-Podolsky-Rosen-Bohm experiments

José L. Cereceda|ArXiv.org|Mar 7, 2000

Quantum Mechanics and Applications15 references7 citations

TL;DR

This paper derives fundamental probabilistic constraints on Einstein-Podolsky-Rosen-Bohm (EPRB) experiments arising from relativistic causality and quantum mechanics. It establishes that while causality restricts joint probabilities via the parameter independence (signal locality) condition, quantum mechanics further constrains these probabilities, leading to a tighter bound of $2\sqrt{2}$ on the CHSH correlation sum—distinct from the general probabilistic maximum of 4. The work also formulates Hardy's nonlocality theorem within the CHSH framework and derives a testable inequality distinguishing quantum mechanics from general probabilistic theories.

ABSTRACT

Relativistic causality, namely, the impossibility of signaling at superluminal speeds, restricts the kinds of correlations which can occur between different parts of a composite physical system. Here we establish the basic restrictions which relativistic causality imposes on the joint probabilities involved in an experiment of the Einstein-Podolsky-Rosen-Bohm type. Quantum mechanics, on the other hand, places further restrictions beyond those required by general considerations like causality and consistency. We illustrate this fact by considering the sum of correlations involved in the CHSH inequality. Within the general framework of the CHSH inequality, we also consider the nonlocality theorem derived by Hardy, and discuss the constraints that relativistic causality, on the one hand, and quantum mechanics, on the other hand, impose on it. Finally, we derive a simple inequality which can be used to test quantum mechanics against general probabilistic theories.

Motivation & Objective

• To identify the general probabilistic constraints imposed by relativistic causality on joint probabilities in EPRB-type experiments.
• To examine how quantum mechanics imposes additional restrictions beyond those from causality alone.
• To analyze the implications of setting three specific joint probabilities to zero, linking it to Hardy's nonlocality theorem.
• To derive a simple inequality that can experimentally test quantum mechanics against general probabilistic theories.

Proposed method

• Derives the causal communication constraint (parameter independence) as a necessary condition for relativistic causality, ensuring measurement outcomes on one particle are independent of distant measurement choices.
• Expresses the CHSH correlation sum $\Delta = c(a_1,b_1) + c(a_1,b_2) + c(a_2,b_1) - c(a_2,b_2)$ in terms of joint probabilities $p(a_j=m, b_k=n)$, using the normalization and causality constraints.
• Applies the non-negativity condition on probabilities to derive further restrictions, particularly in the case where three specific joint probabilities are set to zero.
• Analyzes Hardy's nonlocality scenario within the CHSH framework by assuming $p(a_1=1,b_1=1) = p(a_1=-1,b_2=1) = p(a_2=1,b_1=-1) = 0$, and derives the resulting constraints on the remaining probabilities.
• Derives a testable inequality that discriminates between quantum mechanics and general probabilistic theories by comparing the maximum allowed $\Delta$ under each framework.
• Uses quantum mechanical formalism to show that the probability $p(a_j=m)$ is independent of distant measurement settings, confirming the parameter independence condition.

Experimental results

Research questions

• RQ1 What are the minimal probabilistic constraints imposed by relativistic causality on joint probabilities in EPRB experiments?
• RQ2 How do quantum mechanical probabilities further restrict the CHSH correlation sum beyond the constraints of causality alone?
• RQ3 What constraints arise when three specific joint probabilities are set to zero, and how does this relate to Hardy's nonlocality theorem?
• RQ4 Can a simple inequality be derived to experimentally distinguish quantum mechanics from general probabilistic theories?
• RQ5 What is the maximal possible value of the CHSH correlation sum under general probabilistic theories versus quantum mechanics?

Key findings

• The causal communication constraint ensures that the probability of a measurement outcome on one particle is independent of the distant measurement choice on the other, preserving relativistic causality.
• The CHSH correlation sum $\Delta$ is bounded by 2 in any local hidden variable theory, but quantum mechanics allows a maximum value of $2\sqrt{2}$, which is still below the general probabilistic maximum of 4.
• When three specific joint probabilities are set to zero (as in Hardy's nonlocality), quantum mechanics still allows a non-zero value for the remaining probability, but the sum $\Delta$ is constrained to be less than or equal to $2\sqrt{2}$.
• A general probabilistic theory allows $|\Delta| \leq 4$, but quantum mechanics restricts this to $|\Delta_{\text{QM}}| \leq 2\sqrt{2}$, demonstrating an additional constraint beyond causality.
• The paper derives a testable inequality that can distinguish quantum mechanics from general probabilistic theories by comparing the maximum allowed $\Delta$ values.
• Quantum mechanics satisfies the parameter independence condition, as shown via the quantum formalism: $p(a_j=m) = \langle \psi | \hat{p}_m(\hat{a}_j) | \psi \rangle$, independent of measurements on the distant particle.