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arXiv:1907.01962

Contingent Free Choice: On Extending Quantum Theory to a Contextual, Deterministic Theory With Improved Predictive Power

Ghislain Fourny

Published Jul 3, 2019• 1 citation
arXivHTMLPDF

Abstract

The non-extensibility of quantum theory into a theory with improved predictive power is based on a strong assumption of independent free choice, in which the physicists pick a measurement axis independently of anything that couldn't have been caused by their decision. Independent free choice is also at the core of the Nash equilibrium and classical game theory. A more recent line of game-theoretical research based on weakening free choice leads to non-trivial solution concepts with desirable properties such as at-most uniqueness, Pareto optimality, and contextuality. We show how introducing contingent free choice in the foundations of quantum theory yields a class of deterministic and contextual theories with an improved predictive power, and contrast them with the pilot-wave theory. Specifically, we suggest that quantum experiments, such as the EPR experiment, involving measurements located in spacetime, can be recast as dynamic games with imperfect information involving human agents and the universe. The underlying idea is that a physicist picking a measurement axis and the universe picking a measurement outcome are two faces of the same physical contingency phenomenon. The classical, Nashian resolution of these games based on independent free choice is analogous to local hidden variable theories, constrained by the Bell inequalities. On the other hand, in a setup in which agents are rational and omniscient in all possible worlds, under contingent free choice, the Perfectly Transparent Equilibrium provides a contextual resolution, based on the iterated elimination of inconsistent worlds, towards an at-most unique possible world, in which the outcomes of measurements that actually are carried out, and only them, are deterministically defined.

DOI
10.3929/ethz-b-000351179

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No-Signalling is a fundamental constraint on the probabilistic predictions made by physical theories. It is usually justified in terms of the constraints imposed by special relativity. However, this justification is not as clear-cut as is usually supposed. We shall give a different perspective on this condition by showing an equivalence between No-Signalling and Lambda Independence, or "free choice of measurements", a condition on hidden-variable theories which is needed to make no-go theorems such as Bell's theorem non-trivial. More precisely, we shall show that a probability table describing measurement outcomes is No-Signalling if and only if it can be realized by a Lambda-Independent hidden-variable theory of a particular canonical form, in which the hidden variables correspond to non-contextual deterministic predictions of measurement outcomes. The key proviso which avoids contradiction with Bell's theorem is that we consider hidden-variable theories with signed probability measures over the hidden variables - i.e. negative probabilities. Negative probabilities have often been discussed in the literature on quantum mechanics. We use a result proved previously in "The Sheaf-theoretic Structure of Locality and Contextuality" by Abramsky and Brandenburger, which shows that they give rise to, and indeed characterize, the entire class of No-Signalling behaviours. In the present paper, we put this result in a broader context, which reveals the surprising consequence that the No-Signalling condition is equivalent to the apparently completely different notion of free choice of measurements.

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The non-extensibility of quantum theory into a theory with improved predictive power is based on a strong assumption of independent free choice, in which the physicists pick a measurement axis independently of anything that couldn't have been caused by their decision. Independent free choice is also at the core of the Nash equilibrium and classical game theory. A more recent line of game-theoretical research based on weakening free choice leads to non-trivial solution concepts with desirable properties such as at-most uniqueness, Pareto optimality, and contextuality. We show how introducing contingent free choice in the foundations of quantum theory yields a class of deterministic and contextual theories with an improved predictive power, and contrast them with the pilot-wave theory. Specifically, we suggest that quantum experiments, such as the EPR experiment, involving measurements located in spacetime, can be recast as dynamic games with imperfect information involving human agents and the universe. The underlying idea is that a physicist picking a measurement axis and the universe picking a measurement outcome are two faces of the same physical contingency phenomenon. The classical, Nashian resolution of these games based on independent free choice is analogous to local hidden variable theories, constrained by the Bell inequalities. On the other hand, in a setup in which agents are rational and omniscient in all possible worlds, under contingent free choice, the Perfectly Transparent Equilibrium provides a contextual resolution, based on the iterated elimination of inconsistent worlds, towards an at-most unique possible world, in which the outcomes of measurements that actually are carried out, and only them, are deterministically defined.
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.
We suggest to look at quantum measurement outcomes not through the lens of probability theory, but instead through decision theory. We introduce an original game-theoretical framework, model and algorithmic procedure where measurement scenarios are multiplayer games with a structure all observers agree on. Measurement axes and, newly, measurement outcomes are modeled as decisions with nature being an action-minimizing economic agent. We translate physical notions of causality, correlation, counterfactuals, and contextuality to particular aspects of game theory. We investigate the causal consistency of dynamic games with imperfect information from the quantum perspective and conclude that counterfactual dependencies should be distinguished from causation and correlation as a separate phenomenon of its own. Most significantly, we observe that game theory based on Nash equilibria stands in contradiction with a violation of Bell inequalities. Hence, we propose that quantum physics should be analyzed with non-Nashian game theory, the inner workings of which we demonstrate using our proposed model.
Superdeterminism has received recent attention as a possible path toward a locally causal explanation of the entanglement correlations that appear in experimental tests of Bell's theorem. While the term `superdeterminism' was coined by Bell to refer to restrictions on the free will of experimenters, it was not rigorously defined until recently. It has now been defined as a property of any physical theory that produces systematic violations of statistical independence. Here we focus on formalizing the requirements that being nonsuperdeterministic places on a physical theory, and setting a standard that must be met before we can conclude that a given theory is not superdeterministic. We begin by carefully examining how a physical theory determines what outcomes we observe when performing measurements in terms of ontic states and response functions, and how this differs between superdeterministic and nonsuperdeterministic theories, in terms of the behavior of the types of vetted random sampling procedures that we use in experiments. The core result is that individual samples and measurement outcomes must be representative of the observed distributions, which is explained in detail. This also has a bearing on how measurement settings are chosen by agents, whether freely or randomly, and we argue that this standard ultimately defines what freedom/independence actually mean. We then discuss contextuality, and show that in most cases, superdeterminism is contextual. Finally, we discuss how different physical theories, with different notions of ontic states and response functions, can give rise to the same empirical data, and how the same operational contextuality may appear in different forms.
Abstract We suggest to look at quantum measurement outcomes not through the lens of probability theory, but instead through decision theory. We introduce an original game-theoretical framework, model and algorithmic procedure where measurement scenarios are multiplayer games with a structure all observers agree on. Measurement axes and, newly, measurement outcomes are modeled as decisions with nature being an action-minimizing economic agent. We translate physical notions of causality, correlation, counterfactuals, and contextuality to particular aspects of game theory. We investigate the causal consistency of dynamic games with imperfect information from the quantum perspective and conclude that counterfactual dependencies should be distinguished from causation and correlation as a separate phenomenon of its own. Most significantly, we observe that game theory based on Nash equilibria stands in contradiction with a violation of Bell inequalities. Hence, we propose that quantum physics should be analyzed with non-Nashian game theory, the inner workings of which we demonstrate using our proposed model.
It was argued [1] that there can be no extension of quantum mechanics with improved predictive power on a measurement freely chosen, independently of any event that is not in its future light cone. The assumption of measurement choice was criticized [2] to be too strong to be physically necessary and extensions of quantum mechanics were shown [3] to be possible under a more relaxed measurement assumption. Here I point out an error in the criticism and observe that the actual mistake of the no-go theorem lies in an unwarranted assumption implicitly made in the proof of [1]. Hence, quantum mechanics is guaranteed to have the maximal predictive power only in situations of complete certainty and complete uncertainty about measurement outcomes. I then show that the measurement assumption can be further relaxed without affecting the conclusion on the predictive power of quantum mechanics versus alternative theories. I further study the optimal predicative improvement over quantum mechanics of local spin measurements on a pair of entangled qubits by any alternative theory and conjecture a strict upper bound.
No-Signalling is a fundamental constraint on the probabilistic predictions made by physical theories. It is usually justified in terms of the constraints imposed by special relativity. However, this justification is not as clear-cut as is usually supposed. We shall give a different perspective on this condition by showing an equivalence between No-Signalling and Lambda Independence, or "free choice of measurements", a condition on hidden-variable theories which is needed to make no-go theorems such as Bell's theorem non-trivial. More precisely, we shall show that a probability table describing measurement outcomes is No-Signalling if and only if it can be realized by a Lambda-Independent hidden-variable theory of a particular canonical form, in which the hidden variables correspond to non-contextual deterministic predictions of measurement outcomes. The key proviso which avoids contradiction with Bell's theorem is that we consider hidden-variable theories with signed probability measures over the hidden variables - i.e. negative probabilities. Negative probabilities have often been discussed in the literature on quantum mechanics. We use a result proved previously in "The Sheaf-theoretic Structure of Locality and Contextuality" by Abramsky and Brandenburger, which shows that they give rise to, and indeed characterize, the entire class of No-Signalling behaviours. In the present paper, we put this result in a broader context, which reveals the surprising consequence that the No-Signalling condition is equivalent to the apparently completely different notion of free choice of measurements.