Identifying Quantum Structure in AI Language: Evidence for Evolutionary Convergence of Human and Artificial Cognition

AI summary

Overview

Research area: Quantum cognition applied to artificial intelligence — specifically, testing whether Large Language Models produce probability structures and word distributions that match those previously observed in human cognition and human-written text.

Technical level: Intermediate. The paper uses the mathematical apparatus of quantum cognition (CHSH inequalities, Bose-Einstein and Maxwell-Boltzmann distributions, Hilbert space models), but the experiments themselves are simple question-and-answer tests and word-frequency counts.

Scope (one sentence): The paper reports two sets of cognitive tests — Bell/CHSH inequality tests and word-frequency statistics tests — run on ChatGPT and Gemini as "subjects," and argues that the non-classical results reveal an evolutionary convergence between human and artificial language and cognition.

What This Paper Is About

Quantum cognition research has repeatedly found that human conceptual combinations and language use violate classical (Kolmogorovian) probability and follow quantum-like statistics. This paper asks whether the same non-classical signatures appear when the cognitive agent is not a human but a Large Language Model. The goal is to test LLMs with the same instruments previously applied to humans — the "The Animal Acts" Bell inequality test and the word-distribution statistics of stories — and to interpret any match as evidence of a deeper convergence between human and artificial meaning-making.

Key Contributions

1. An LLM version of the CHSH/Bell inequality test. The authors administered the same "The Animal Acts" conceptual-combination questionnaire used with 81 human participants (Aerts & Sozzo, 2011) to ChatGPT and Gemini, under three different procedures (guided conversation, an exploratory prompt, and a neutral prompt), and found systematic violations of the CHSH inequality in every case.

2. A statistical analysis of LLM-generated narratives. Stories written at the authors' request by ChatGPT and Gemini — including a Winnie the Pooh story in the style of A. A. Milne and an H. G. Wells-style story by Gemini — were analyzed for word-frequency distribution, and exhibited Bose-Einstein rather than Maxwell-Boltzmann clustering.

3. A parallel between artificial and human results. The paper explicitly frames the LLM results as mirroring earlier results from human cognitive tests and from information-retrieval tests on large human-written corpora.

4. A proposed explanatory hypothesis and unifying framework. The authors argue that the meaning-bearing structure is the distributive semantic structure of vector spaces built on top of the neural network, not the neural network itself, and that this structure is what enables convergence between biological and artificial cognition. A unifying framework linking human language, AI language, and quanta of light is advanced in Section 4 (the truncated content indicates this section by reference, without the full analysis).

Main Findings

• Human baseline: The earlier human experiment with 81 participants produced expectation values E(A,B) = −0.7778, E(A′,B) = 0.6543, E(A,B′) = 0.3580, E(A′,B′) = 0.6296, giving CHSH = 2.4197 — above the classical bound of 2 and below the quantum maximum (Cirel'son's bound) of 2√2 ≈ 2.8284. A single-sample t-test against the value 2 gave p = 0.0171, below 0.05.

• Maximum violation under guided conversation: When ChatGPT and Gemini were walked step by step through the questions in a conversation, they produced E(A,B) = −1 and E(A′,B) = E(A,B′) = E(A′,B′) = 1, giving a maximal CHSH value of 4.

• Non-maximal violations under repeated trials: When asked to perform the four measurements 81 times over 8 different sessions (matching the human sample size), ChatGPT produced CHSH = 2.25 under the exploratory 'prompt 1' (E(A,B) = −0.802, E(A′,B) = 0.235, E(A,B′) = 0.284, E(A′,B′) = 0.926) and CHSH = 3.09 under the neutral 'prompt 2' (E(A,B) = −0.852, E(A′,B) = 0.556, E(A,B′) = 0.679, E(A′,B′) = 1.000).

• Gemini results: Gemini gave CHSH = 2.12 for 'prompt 1' and a maximal CHSH = 4 for 'prompt 2'. Averaging the two prompts gives 2.67 for ChatGPT and 3.06 for Gemini.

• Interpretation of the violations: Following Itamar Pitowsky, the authors state that a CHSH violation proves the underlying probability model is non-Kolmogorovian — events cannot be embedded in a single global probability space with a joint distribution. They connect this to Boole's 1854 "conditions of possible experience" and to Schrödinger's entanglement concept, describing the phenomenon as words behaving differently in combination than in isolation.

• Bose-Einstein word statistics in AI-written stories: In Gemini's Winnie the Pooh story, the most frequent word, the indefinite article "A", appears 132 times and was assigned the lowest energy level E₁; the second most frequent word, "The", appears 98 times. The analysis yielded 822 energy levels for the story.

• Model comparison: Assigning each word an energy level by frequency rank (E_i = i − 1) and fixing two constraints — total word count and total "energy" of the text — the Bose-Einstein distribution fit the story data with great precision, while the Maxwell-Boltzmann distribution, derived from the identical constraints, diverged significantly and "fails both quantitatively and qualitatively."

• Notable qualitative observations: Both LLMs initially claimed writing the story would take a few days and then failed to deliver — which the authors describe as a hallucination (AIs have no internal sense of time) plus possible copyright restrictions, tool malfunctions, or other systemic limits. The authors also report asking for the Winnie the Pooh story specifically because a human-written Winnie the Pooh story was the text in which they had first discovered Bose-Einstein statistics. The three resulting stories are described as "beautiful and well-written."

• Proposed mechanism: The authors propose that meaning itself plays the role that quantum coherence plays for physical bosons, creating a "semantic condensation" analogous to a Bose-Einstein condensate, and that meaning — not grammar — drives the statistical structure, since the AI generates sentences based on meaning patterns learned in training rather than grammatical rules.

• Not reported: The truncated content does not include the full Section 4 unification analysis, the full statistical fit parameters (the fitted values of the constants A, B, C, D, or goodness-of-fit metrics beyond qualitative descriptions), the complete Appendix A–C material, or the full detail of the energy-level spacing exponent d introduced in Eq. 7.

Methodology in Plain English

The authors treat LLMs as experimental subjects and reuse instruments designed for humans.

Test 1 — Conceptual combinations. They take the sentence "The Animal Acts" and decompose it into the concept Animal (with exemplars Horse/Bear and Tiger/Cat) and the concept Acts (with exemplars Growls/Whinnies and Snorts/Meows). Subjects are shown four "coincidence" question options — for instance "The Horse Growls", "The Bear Whinnies", "The Horse Whinnies", "The Bear Growls" — and their choices are converted into +1/−1 outcomes. From the choice frequencies, the researchers compute four expectation values and combine them into the CHSH expression. If the result exceeds 2, the classical probability assumption is violated. They ran this three ways: a guided step-by-step conversation (Appendix A), an "exploratory" prompt inviting imagination and creativity (prompt 1, Appendix B), and a neutral prompt with that instruction omitted (prompt 2, Appendix B). The prompts were repeated 81 times across 8 sessions to match the human sample size.

Test 2 — Word statistics in stories. They asked ChatGPT and Gemini to write stories (a Winnie the Pooh story in A. A. Milne's style, plus an H. G. Wells-style story from Gemini) with no content guidance. They then counted word frequencies, assigned the most frequent word energy level 0, the next energy level 1, and so on, and fixed the total number of words and total energy to the values in the actual story. Those two constraints uniquely determine both a Bose-Einstein curve and a Maxwell-Boltzmann curve. They then checked which curve matched the observed rank-frequency data.

Conceptual framing. The authors interpret violations of the CHSH inequality via Pitowsky's correlation-polytope argument as proof of a non-Kolmogorovian probability model, and they distinguish physical non-locality (no-signaling in spacetime) from the semantic version at work here, where the "non-locality" is structural organization within a shared, high-dimensional semantic field. They borrow the term "cogniton" from their earlier work to name the quantum-like unit of meaning, with a word in a story being a "state" of that cogniton, analogous to a photon of a given frequency or an atom at a given energy level.

Why This Matters

Impact on research. The paper argues that quantum cognition's models — developed for human decision anomalies — transfer directly to LLMs, and it reframes what LLMs are. The authors state that although LLMs are classified as neural networks "for historical reasons," they believe the essential form of knowledge organization is the distributive semantic structure of the vector spaces built on top of the network. If correct, this shifts attention from architecture to meaning representation and suggests a common mathematical description of human and artificial conceptual combination.

Real-world applications (extrapolated from the paper's framing; the paper itself does not report deployed applications):

• Evaluating generative models for hallucination and failure modes. The paper's account of the LLMs promising a story "in a few days" and then silently failing is presented as a concrete example of hallucination and silent termination that evaluation tooling would need to catch.
• Semantic search and information retrieval. The paper's earlier information-retrieval tests on large corpora and its word-statistics analysis bear directly on how term frequency distributions and meaning clustering are modeled in retrieval systems.
• Cross-domain translation between human and machine language. A shared non-classical structure for human and AI language is relevant to any system that must align, compare, or translate between the two.
• Foundational questions for AI safety and interpretability. If meaning organization has quantum-like features such as entanglement-like non-separability of concepts, then attempts to decompose model behavior into independent parts may face principled limits.

Industry relevance. The results are relevant to teams building or auditing language models, particularly those working on prompting behavior (the paper shows the prompt changes the measured violation: CHSH 2.25 versus 3.09 for ChatGPT, 2.12 versus 4 for Gemini), on text-statistics-based detection of machine-generated content, and on the theoretical foundations of vector-space semantic representations.

Future Directions

1. Extend the CHSH and Boole-condition tests to more LLMs and more conceptual combinations. The paper tested ChatGPT and Gemini; the authors explicitly note that other Boole-derived "conditions of possible experience" have been tested in humans (Bruza et al., 2023) and could be applied to artificial agents.

2. Establish whether prompt framing systematically controls the degree of non-classicality. The two prompts produced measurably different CHSH values in both models, raising the question of what determines the strength of the violation and whether violations closer to the human range can be produced reliably.

3. Develop and test the unifying framework linking human language, AI language, and quanta of light. Section 4 of the paper introduces this program, motivated by Martin Klein's observation that Einstein sought to unify Maxwell's electromagnetism with Boltzmann's thermodynamics; the full quantitative development is not contained in the truncated content.

4. Apply the energy-level formalism more broadly. The paper introduces a power d in the energy-level spacing formula E_i = (i−1)^d, motivated by analogies with the harmonic oscillator, the hydrogen atom (i^−2 dependence), and the particle-in-a-box (+2 power). Determining which spacing law best describes language, and why, remains an open question in the truncated content.

Target Audience

Researchers in quantum cognition and cognitive science interested in whether their models extend beyond humans; NLP and machine learning researchers curious about non-classical structure in model outputs; philosophers and theorists of AI working on the relationship between meaning, vector spaces, and neural architectures; and quantitative linguists working on word-frequency distributions and their deviations from classical statistics. Some familiarity with probability theory and basic quantum-mechanical notation is helpful, but the experimental design is accessible to a general technical reader.

Authors’ abstract

We present the results of cognitive tests on conceptual combinations, performed using specific Large Language Models (LLMs) as test subjects. In the first test, performed with ChatGPT and Gemini, we show that Bell's inequalities are significantly violated, which indicates the presence of a 'non-classical probability model' with probabilities that do not satisfy Kolmogorov's axioms. In the second test, also performed using ChatGPT and Gemini, we identify the presence of 'Bose-Einstein statistics', rather than the intuitively expected 'Maxwell-Boltzmann statistics', in the distribution of the words contained in large-size texts. Interestingly, these findings mirror the results previously obtained in both cognitive tests with human participants and information retrieval tests on large corpora. Taken together, they point to the 'systematic emergence of non-classical quantum-like structures in conceptual-linguistic domains', regardless of whether the cognitive agent is human or artificial. Although LLMs are classified as neural networks for historical reasons, we believe that a more essential form of knowledge organization takes place in the distributive semantic structure of vector spaces built on top of the neural network. It is this meaning-bearing structure that lends itself to a phenomenon of evolutionary convergence between human cognition and language, slowly established through biological evolution, and LLM cognition and language, emerging much more rapidly as a result of self-learning and training. We analyze various aspects and examples that contain evidence supporting the above hypothesis. We also advance a unifying framework that explains the pervasive quantum organization of meaning that we identify.