**In 1935, Einstein and two of his peers devised a famous thought experiment which they believed exposed the incompleteness of quantum theory, suggesting that its probabilistic pronouncements represent our uncertainty regarding further aspects of atomic processes not yet formalised. After contested discussion, in the 1960s the defiance of Bell’s inequality was proposed as an enigma that counters this view and has gone largely unopposed since. In three recently published papers, Dr Frank Lad, at the University of Canterbury in New Zealand, identifies a mathematical error in this proposition, pertinent to three seminal papers of quantum theory. His analysis is controversial, but if confirmed the results should lead physicists to reconsider Einstein’s long-dismissed concept of ‘local realism’.**

One of the most celebrated aspects of Einstein’s mind was his ability to construct simple yet deeply meaningful thought experiments. Whether imagining what a person would see if they were riding a light beam and observing another beam travelling in the opposite direction, or what they would weigh when shut inside a closed box in freefall, these ideas were a key step towards his eventual theories of general relativity. Yet another highly influential thought experiment came 20 years later – this time, regarding the physical descriptions of reality provided by the newly formed theory of quantum mechanics.

The results of quantum theory, which Einstein supported and helped to formalise, provide that the activity of an atomic particle monitored at one observation station may depend on the activity of a similar particle travelling toward another station some two kilometres away. Apparently contrary to Einstein’s theory of relativity, this dependence can pertain even if there is no way that information can be transmitted from one station to the other without exceeding the speed of light. Initial considerations involved experimentation detecting magnetic spins of paired electrons, but subsequent developments involved the polarisation of paired photons. What could account for results of quantum entanglement, which Einstein famously characterised as ‘spooky action at a distance’?

In collaboration with two other notable physicists, Boris Podolsky and Nathan Rosen, Einstein proposed that the deliberations of quantum theory must be incomplete, publishing a controversial paper in 1935 which became known by their initials (EPR).

### The EPR debate

Considered as a complete theory, quantum mechanics proposes that observable quantum behaviour is governed only by the probabilities which it identifies.

In their paper, EPR assessed a particular paired experimental setup in which magnets at the two stations are positioned as either aligned or as opposed to one another. From quantum equations, it appeared that the observation of activity at one station must determine the behaviour occurring at the other, ensuring the ‘reality’ of the condition. The EPR proposal that the theory is incomplete called for the recognition of further unspecified conditions of the general physical process that are not yet incorporated in the theory. Although the theory may describe behaviour of an ensemble of many observations, the quantum probabilities for individual-paired outcomes must represent scientific uncertainty regarding the status of these supplementary conditions. Einstein wrote in a famous letter to a friend that he could ‘not believe that the old one rolls dice’. The EPR proposition was meant to preserve the principle of ‘local realism,’ in keeping with Einstein’s theory of general relativity. This presumes that no event can be triggered by another if the distance separating them cannot be covered by light within the time separating them.

“Dr Lad has shown that quantum theory doesn’t support the expectation value of 2√2. It only specifies an interval for E(s) within 1.1213 and 2, because the four relevant polarisation products inhere neglected symmetric functional relations.”

These ideas conflicted with a foundational principle in quantum mechanics, which specifies that the second particle in an entangled pair could have any state before the first is measured. It is understood that the states of the dual pair are ‘superposed’. In contrast, the local realism demanded by the EPR paradox proposed that entangled particles are instead like pairs of separated gloves: the observation of one will tell whether the other is left- or right-handed – but will never change the glove’s handedness.

The paper generated a flurry of constructive discussions in which many leading figures of the day participated. The conclusion widely accepted was that the quantum theorists had discovered a remarkable result: at its smallest scale physical activity is random, and the theory had identified a natural tendency in matter itself towards specific probabilistic outcomes. Mathematical research at the time was actively directed to the ‘laws of large numbers’, which are considered to support the regularity of random experimental observations. However, with no definitive argument, debate continued over whether the measurement outcomes of the particles really are probabilistic in themselves.

### Enter John Bell and his inequality

Debate continued after the trio’s initial proposal, without a conclusive argument emerging. Yet everything changed in 1964, when physicist John Stewart Bell came up with a thought experiment of his own. Bell had begun his investigations with a mind to reconfirming Einstein’s position. However, he came upon a puzzling result that startled the world of theoretical quantum physics. Eventually, it appeared to be confirmed by optical experimentation.

In an optical setting, Bell’s considerations involve a pair of photons for which the polarisers they meet are neither parallel nor perpendicular. In this case, he found that if the principle of local realism is presumed to hold then the probabilistic requirements of quantum theory must defy a simple inequality of probability. This was subsequently named after him, as ‘Bell’s inequality.’ It appeared that in the new physics, the principle of local realism would need to be abandoned at the quantum scale.

The thought experiment involves the impossible simultaneous transmission of a single pair of photons to four different polariser angle pairings, with the numerical product of polarisation observations recorded at all of them. The critical quantity relevant to the defiance of Bell’s inequality is specified as a linear combination of all four of them, labelled ’s’. Numerically, each component product is limited to an observation value of either -1 or +1. Although s might conceivably be as small as -4 or as large as +4, the principle of local realism would allow only values of either -2 or +2. Although the value of s in the gedankenexperiment is recognised as random, its expectation E(s) should surely lie within the interval [-2,+2]. This is the simple statement of Bell’s inequality in this context: |E(s)| <= +2.

In defiance, quantum analysis identifies the expected value of any single polarisation product in the context of Bell’s thought experiment as 1/√2. Therefore, the expectation of the linear combination of the four products appears to equal 4/√2, or 2√2, a value which clearly exceeds +2. In contrast with the mechanics of classical scales which presume the validity of local realism, the defiance of Bell’s inequality made the physics of the quantum world appear to be nothing short of absurd. There are apparently mysterious connections between the components of matter at the quantum scale, which defy our naïve notions of reality at macroscopic scales. In the words of N David Mermin, one of the leading figures of 20th-century physics, the mystery is the behaviour ‘of the atomic world itself, acting at its most perverse’.

### Empirical investigations

From the very beginning of his studies of the matter, Bell wasn’t fully convinced that quantum theory could really offer a full mathematical description of quantum entanglement. Though he could not identify the error involved in inequality defiance, he suspected that someday it would be discovered. However, this didn’t prevent his results from taking the wider quantum physics community by storm in the coming decades.

“Dr Lad argues that his results provide a clear demonstration of the error which Bell suspected, but ultimately wasn’t able to find. “

Surprised that such a seminal result of quantum physics could rely on a ‘thought experiment’, Alain Aspect, a young French physicist specialising in optics, devised a pertinent physical experiment. He studied polarisation product observations from sequences of paired photons at each of the four relative angle pairings, designed to estimate their expectations. The result at each angle pairing yielded an estimate near to 1/√2 to several decimal places. Astonishing the research field, he dutifully pronounced an empirical confirmation of the defiance of Bell’s inequality. Although several loopholes were subsequently considered to qualify Aspect’s results, they have all since been debunked.

Theoretical investigations seemed to be even more decisive. In 1991, a landmark study by physicists Daniel Greenberger, Michael Horne, Abner Shimony, and Anton Zeilinger (GHSZ) took the idea a step further. Their seminal paper claimed that the very formulation of models involving local realism entails unavoidable logical contradiction. Support for local realism at a quantum scale was on the wane.

### Uncovering Bell’s error

Although the defiance of Bell’s inequality has now become mainstream dogma in theoretical physics, some researchers have remained sceptical. From the early days of the discussion, the philosopher Arthur Fine realised that quantum theory motivates no joint distribution for the four components of s. The general uncertainty principle denies the prospect of making assertions regarding the joint outcomes of simultaneous experiments that cannot be conducted. Analysis of the thought experiment continued in minority circles despite repeated rebuttal by quantum theorists.

In three recent publications, Dr Frank Lad at the University of Canterbury in Otautahi, New Zealand, claims to have discovered the mathematical error that John Bell had suspected in his analysis. In the *Journal of Modern Physics (JMP),* Lad reported that the Aspect/Bell calculations neglect the relevance of four symmetric functional restrictions among the components of s in the thought experiment. These are critical for the assessment of its expected value. He has shown that quantum theory doesn’t support the expectation value of 2√2 at all. Rather, it only specifies an interval for E(s) within 1.1213 and 2. Moreover, a simulation study using the quantum probabilities in a believable scenario yields a value of 1.7668, though this value cannot be definitive.

Furthermore, Dr Lad has identified in the journal *Entropy* that the analysis proposed by GHSZ is based on a self-contradictory assumption that these investigators had introduced into their analysis themselves. Finally, in the third article in *JMP*, he found that a celebrated exposition of the quantum mysteries by Professor David Mermin of Cornell University is based upon a mistake that confuses the conduct of a real experiment and a thought experiment.

Dr Lad argues that his analysis provides a clear demonstration of the error which Bell suspected but ultimately wasn’t able to find. If correct, this conclusion could be monumentally significant.

### Understanding quantum mechanics and resolving controversy

Lad’s results do not deny any real probabilistic forecasts of quantum theory itself, nor the validity of any experiment that has been conducted over the past half-century. ‘They merely challenge the analysis of Bell’s thought experiment and the inferences that have been purported to derive from experimental data’, he says. ‘The dons of theoretical quantum physics have long sheltered their claims to non-locality from substantive dispute by protecting their journals from formal analysis that exposes them as mistaken. This new challenge needs to be addressed by reasoned technical assessment of its argument rather than by tight-lipped dismissal.’

Clearly, Dr Lad’s conviction in the validity of Bell’s inequality and its coherence with locality places him in disagreement with many previous studies, whose results have been widely accepted in the quantum physics community. Ultimately, physicists are by now well aware that several aspects of quantum mechanics in its current form are incomplete: from the need to unify the phenomena with the gravity described by Einstein’s general relativity, to long-awaited updates for the Standard Model of particle physics. The fundamental forces and particles currently described therein cannot explain phenomena such as dark matter or dark energy. Through such a clear break with the presumption of non-locality in quantum behaviour which has long gone undisputed, Dr Lad’s ideas may one day set physicists on a path to answering these questions.

What inspired you to conduct this research?

What inspired you to conduct this research?

A subjective probabilist in the mode of Bruno de Finetti’s thought, I had long been disinterested by problems of physics, and was offended by touted claims to the discovery of randomness inhering in Nature. In this state I heard from a probabilist friend that the defiance of Bell’s inequality was a matter well worthy of investigation, and he led me into it.

**References**

- Lad, F, (2021). Quantum Mysteries for No One.
*Journal of Modern Physics,*12, 1366. doi.org/10.4236/jmp.2021.129082 - Lad, F, (2021). Quantum Violation of Bell’s Inequality: a Misunderstanding based on a Mathematical Error of Neglect.
*Journal of Modern Physics,*12, 1109. doi.org/10.4236/jmp.2021.128067 - Kupczynski, M, (2020). Is the Moon There if Nobody Looks: Bell Inequalities and Physical Reality.
*Frontiers in Physics,*8, 273. doi.org/10.3389/fphy.2020.00273 - Lad, F, (2020). The GHSZ Argument: A Gedankenexperiment Requiring More Denken.
*Entropy,*22, 759. doi.org/10.3390/e22070759 - Wiseman, H, (2015). Death by Experiment for the Principle of Local Realism.
*Nature,*526, 649. doi.org/10.1038/nature15631

**DOI**

10.26904/RF-138-1808783260

### Research Objectives

In his collection of papers, Frank Lad claims to have discovered a mathematical error regarding the defiance of Bell’s inequality.

### Bio

Frank Lad is a Senior Research Fellow in the Department of Mathematics and Statistics, University of Canterbury, Otautahi/Christchurch, New Zealand. He is author of a pedagogical text exposing the thought of Bruno de Finetti: Operational Subjective Statistical Methods: a mathematical, philosophical, and historical introduction, New York: John Wiley, 1996.

**Contact**

Frank Lad

337 Worcester Street

Linwood

Otautahi/Christchurch

New Zealand, 8011

E: frank.lad@canterbury.ac.nz

T: +64 3 365 1789

W: www.researchgate.net/profile/Frank-Lad