Extending quantum theory with AI-assisted deterministic game theory

We present an AI-assisted framework for predicting individual runs of complex quantum experiments, including contextuality and causality (adaptive measurements), within our long-term programme of discovering a local hidden-variable theory that extends quantum theory. In order to circumvent impossibility theorems, we replace the assumption of free choice (measurement independence and parameter independence) with a weaker, compatibilistic version called contingent free choice. Our framework is based on interpreting complex quantum experiments as a Chess-like game between observers and the universe, which is seen as an economic agent minimizing action. The game structures corresponding to generic experiments such as fixed-causal-order process matrices or causal contextuality scenarios, together with a deterministic non-Nashian resolution algorithm that abandons unilateral deviation assumptions (free choice) and assumes Perfect Prediction instead, were described in previous work. In this new research, we learn the reward functions of the game, which contain a hidden variable, using neural networks. The cost function is the Kullback-Leibler divergence between the frequency histograms obtained through many deterministic runs of the game and the predictions of the extended Born rule. Using our framework on the specific case of the EPR 2-2-2 experiment acts as a proof-of-concept and a toy local-realist hidden-variable model that non-Nashian quantum theory is a promising avenue towards a local hidden-variable theory. Our framework constitutes a solid foundation, which can be further expanded in order to fully discover a complete quantum theory.

💡 Research Summary

The paper proposes a novel framework that recasts quantum experiments as deterministic extensive‑form games between observers and the universe, and then uses artificial intelligence to learn the hidden‑variable reward functions that reproduce quantum statistics. The authors begin by revisiting the standard impossibility theorems—Bell’s inequality, the Kochen‑Specker theorem, the Free‑Will theorem, etc.—which all rely on a strong free‑choice assumption (measurement independence together with parameter independence). They argue that this assumption is the only loophole left for a local‑realist hidden‑variable model and replace it with a weaker, compatibilist notion called “contingent free choice.”

In their game‑theoretic representation, each quantum protocol (including fixed‑order process matrices and causal‑contextuality scenarios) is translated into an extensive‑form game with imperfect information. The players are the observers and the universe (or particles) and the universe is modeled as an economic agent that minimizes an action‑like quantity. Traditional Nash equilibrium requires unilateral deviation—i.e., free choice—and therefore cannot reproduce quantum correlations within this deterministic setting. To overcome this, the authors introduce the Perfect Prediction Equilibrium (PTE), a non‑Nashian solution concept in which all players perfectly predict each other’s strategies. PTE drops the unilateral deviation axiom, assumes perfect mutual prediction, and when it exists it is unique and Pareto‑optimal. This matches the “contingent free choice” stance: observers are not free in the strong sense but are not fully super‑deterministic either.

The central technical contribution of the present work is the learning of the reward functions that assign a payoff to every possible history (a combination of measurement settings and outcomes). These reward functions are parameterized by a neural network, optionally constrained by physics‑inspired architectural choices (e.g., symmetry under rotations). A differentiable PTE solver is built: the solver takes the neural‑network‑generated rewards, computes the equilibrium strategies for a large sample of hidden variables λ, and produces a histogram of simulated outcomes. To make the solver amenable to gradient‑based training, the authors introduce an annealed decision temperature, which smooths the discrete decision process and yields a differentiable loss landscape.

Training proceeds by minimizing the Kullback‑Leibler (KL) divergence between two distributions: (i) the empirical histogram obtained from many deterministic game runs (each run uses a randomly drawn λ) and (ii) the quantum‑theoretic distribution given by the extended Born rule for the same experimental protocol (derived from the process matrix and the chosen quantum instruments). Stochastic gradient descent updates the neural‑network parameters until the KL divergence is minimized, meaning the learned rewards generate game outcomes that match quantum predictions.

The authors demonstrate the method on the canonical 2‑2‑2 Einstein‑Podolsky‑Rosen (EPR) scenario (Bohm’s spin version). The game structure is a two‑player extensive‑form tree with imperfect information; the learned reward function collapses effectively to a single parameter—the angular offset between the two measurement bases—consistent with the known quantum solution. Simulations with millions of deterministic runs reproduce the quantum probability distribution to high accuracy, and the resulting statistics violate the Bell inequality despite the underlying deterministic game dynamics. The authors note two sources of residual discrepancy: finite sample size and incomplete convergence of the neural network within the allotted training time.

Beyond the proof‑of‑concept, the paper discusses how the same pipeline could be applied to more complex experiments (multi‑party, higher‑dimensional settings) where analytic reward ansätze are unavailable. Potential extensions include symbolic regression for interpretable reward forms, richer neural architectures, and incorporation of additional physical constraints such as the principle of least action for spin measurements.

In conclusion, the work offers a concrete route to embed local hidden‑variable models within a deterministic game‑theoretic framework that respects a weakened free‑choice assumption and reproduces quantum statistics via AI‑driven learning of hidden‑variable reward functions. While still preliminary—particularly regarding scalability and the physical interpretation of the learned rewards—it opens a promising interdisciplinary avenue linking quantum foundations, game theory, and machine learning, and suggests that non‑Nashian equilibria may provide a viable loophole to the traditional no‑go theorems.