[Paper Review] Exclusion of Time in Mermin's Proof of Bell-Type Inequalities

K. Hess, Walter Philipp|ArXiv.org|May 7, 2003

History and advancements in chemistry3 references3 citations

TL;DR

This paper challenges Mermin's nontechnical proof of Bell-type inequalities by demonstrating that his argument fails when time and setting-dependent instrument parameters are properly accounted for in the hidden variable model. The authors show that Mermin's conclusion—that quantum correlations can be explained classically—collapses under realistic stochastic dependencies, invalidating his proof in extended parameter spaces.

ABSTRACT

Mermin states that his nontechnical version of Bell's theorem stands and is not invalidated by time and setting dependent instrument parameters as claimed in one of our previous papers. We identify deviations from well-established protocol in probability theory as well as mathematical contradictions in Mermin's argument and show that Mermin's conclusions are therefore not valid: his proof does not go forward if certain possible time dependencies are taken into account.

Motivation & Objective

• To refute Mermin's claim that his nontechnical proof of Bell-type inequalities remains valid despite time and setting-dependent instrument parameters.
• To identify mathematical and probabilistic inconsistencies in Mermin's argument when extended to include time-dependent hidden variables.
• To demonstrate that Mermin's proof fails in a broader parameter space that includes stochastic dependencies between instrument parameters and the source.
• To counter Mermin's assertion that quantum mechanics implies time-invariant instrument settings, showing such constraints are physically unjustified and mathematically unnecessary.
• To establish that the extended parameter space used in the authors' prior work does not reduce to Mermin's model, thereby preserving the validity of their earlier critique of Bell-type theorems.

Proposed method

• Reformulates Mermin's proof using formal probability notation, replacing his 'instruction sets' with random variables $A, B = \pm 1$ indexed by settings $\mathbf{a}, \mathbf{b}, \mathbf{c}$.
• Introduces time and setting-dependent instrument parameters $\lambda_i^*(t)$ and $\lambda_j^{**}(t)$, allowing arbitrary stochastic dependencies from the source parameter $\Lambda$.
• Constructs a model where the joint probability density of instrument parameters depends on both time and settings, using arbitrary algorithms to determine parameter order.
• Applies the condition $A_{\mathbf{i}} = B_{\mathbf{i}}$ for identical settings (from Mermin's feature i), and derives the row sum $\sum_{\text{row}}(\lambda_s) = (A_{\mathbf{a}} + A_{\mathbf{b}} + A_{\mathbf{c}})(B_{\mathbf{a}} + B_{\mathbf{b}} + B_{\mathbf{c}})$, which equals 1 or 9.
• Shows that Mermin's average row sum $\geq 1/9$ is invalid under time-dependent instrument parameters, as the assumption of stochastic independence breaks down.
• Demonstrates that time shifts $\Delta_t$ in measurements can be modeled via causal transformations of time-dependent functions, refuting claims of 'spooky conspiracy' in the model.

Experimental results

Research questions

• RQ1 Does Mermin's nontechnical proof of Bell-type inequalities remain valid when instrument parameters depend on time and measurement settings?
• RQ2 Can a hidden variable model with time- and setting-dependent instrument parameters still satisfy the statistical features of quantum mechanics as described by Mermin?
• RQ3 Is the stochastic independence between instrument parameters and the source parameter preserved in Mermin's model under extended parameter spaces?
• RQ4 Does the assumption of time-invariant instrument settings, as claimed by Mermin, have a basis in quantum mechanics or in experimental feasibility?
• RQ5 Can a causal, time-dependent model reproduce the quantum mechanical predictions without requiring nonlocal or conspiratorial correlations?

Key findings

• Mermin's proof fails when time and setting-dependent instrument parameters are introduced, as his assumption of stochastic independence between instrument parameters and the source breaks down.
• The extended parameter space used in the authors' prior work does not collapse onto Mermin's model, invalidating his claim that their critique is irrelevant.
• The average row sum in Mermin's table, which he claims is $\geq 1/9$, becomes invalid under time-dependent instrument parameters, as the derivation relies on unjustified independence assumptions.
• Mermin's argument that quantum mechanics implies time-invariant instrument settings is not supported by the mathematics of Bell-type theorems and is experimentally unverifiable over long time intervals.
• Time shifts $\Delta_t$ in measurements can be modeled causally via time-dependent functions, showing that apparent 'conspiracies' are merely standard cause-effect relationships.
• The authors' model can reproduce quantum mechanical predictions without requiring nonlocal or 'spooky' correlations, refuting Mermin's claim of conspiratorial elements.