[Paper Review] Neural networks as "hidden" variable models for quantum systems

Steven J. Weinstein|arXiv (Cornell University)|Jul 11, 2018

Quantum Mechanics and Applications29 references4 citations

TL;DR

This paper proposes conditional Restricted Boltzmann Machines (cRBMs) as hidden variable models for quantum two-spin systems, successfully reproducing product and entangled states—including the EPR-Bohm singlet state—by making the system state depend on both preparation and measurement settings. The model circumvents Bell's theorem by introducing contextuality that mimics retrocausal effects, with conceptual parallels in Wheeler-Feynman absorber theory and electric guitar circuits.

ABSTRACT

We successfully model the behavior of two-spin systems using neural networks known as conditional Restricted Boltzmann Machines (cRBMs). The result gives local hidden variable models for product states and entangled states, including the singlet state used in the EPR-Bohm experiment. Bell's theorem is circumvented because the state of the system is dependent not only on the preparation but also on the measurement setup (the detector settings). Though at first glance counterintuitive, the apparent retrocausality in these models has a historical precedent in the absorber theory of Wheeler and Feynman and an intuitive analog in the simple AC circuit of an electric guitar.

Motivation & Objective

• To develop a classical machine learning model capable of reproducing quantum correlations in two-spin systems, including entangled states.
• To explore whether neural networks can serve as hidden variable models that reproduce quantum behavior without violating Bell's theorem.
• To investigate how contextuality—dependence on measurement settings—can be encoded in a neural network to simulate quantum-like correlations.
• To provide a classical, intuitive framework for understanding nonlocal quantum correlations through a physically motivated model inspired by absorber theory and AC circuits.

Proposed method

• The study employs conditional Restricted Boltzmann Machines (cRBMs), a type of deep generative model, to represent the joint probability distribution of measurement outcomes in two-spin systems.
• The cRBM is trained to model the conditional probabilities of measurement outcomes given specific detector settings, effectively encoding contextuality.
• The model explicitly incorporates measurement settings as inputs, making the system state dependent on both preparation and measurement context.
• The architecture allows the network to learn and reproduce both product states and entangled states, including the singlet state.
• The model’s structure is motivated by the absorber theory of Wheeler and Feynman, which introduces retrocausal effects in classical electrodynamics.
• An intuitive analogy is drawn to the AC circuit of an electric guitar, where feedback effects resemble the context-dependent behavior of the model.

Experimental results

Research questions

• RQ1 Can a classical neural network model reproduce quantum correlations in two-spin systems without violating Bell's theorem?
• RQ2 How can contextuality—dependence on measurement settings—be encoded in a generative model to simulate quantum entanglement?
• RQ3 What is the physical and conceptual basis for the apparent retrocausality in the model, and how does it relate to established physical theories?
• RQ4 Can the behavior of the model be understood through analogies in classical physics, such as absorber theory or electric guitar circuits?

Key findings

• The cRBM successfully models both product states and entangled states, including the EPR-Bohm singlet state, with high fidelity.
• The model reproduces quantum correlations without violating Bell's theorem by making the system state dependent on measurement settings, thus introducing contextuality.
• The apparent retrocausality in the model finds precedent in the absorber theory of Wheeler and Feynman, which also features time-symmetric interactions.
• The model’s behavior is intuitively analogous to the feedback mechanism in an electric guitar’s pickup circuit, where the output depends on the entire system, including future conditions.
• The neural network’s ability to encode contextuality provides a classical framework for understanding quantum nonlocality.
• The results suggest that quantum-like correlations can emerge from classical, context-dependent models when measurement settings are explicitly included in the state description.