Beyond Hidden Variables: The Limits of Local Realism in Quantum Dynamics

Author: Denis Avetisyan

A new analysis probes whether the evolution of quantum systems can be fully explained by hidden variables, revealing fundamental constraints on local realism.

If the time evolution of quantum measurement statistics-the probabilities $P(a|x,\rho(t))$ of observing a specific outcome upon measurement-can be fully reproduced by local hidden variable (LHV) models given a subset of local quantum states within the broader quantum state space, then the dynamics suggest a potential underlying mechanism where hidden variables, rather than inherently non-local quantum effects, dictate the observed probabilities.

This review examines the dynamics of local hidden-variable models and demonstrates their inability to fully describe the time evolution of interacting quantum systems.

While Bell’s theorem establishes limits on static correlations within quantum mechanics, the role of time evolution in local realism remains largely unexplored. This motivates the study presented in ‘On the Dynamics of Local Hidden-Variable Models’, which investigates whether the temporal development of quantum correlations can be consistently described by evolving local hidden variables. Our analysis suggests that such a description is not generally possible, particularly in interacting quantum systems, revealing a novel form of nonlocality linked to dynamical processes. Could this dynamic nonlocality offer new insights into the foundations of quantum mechanics and its implications for information processing?

Unveiling Classical Foundations: The Illusion of Local Realism

The foundation of classical physics rests upon the principle of local realism, a worldview where objects possess definite properties independent of measurement and where any influence between objects is limited by the speed of light. This implies that a particle’s characteristics, such as its position or momentum, are not merely probabilities until observed, but pre-existing values. Furthermore, local realism dictates that an object can only be directly influenced by its immediate surroundings; distant events cannot instantaneously alter its state. This concept of locality is crucial, as it rejects the possibility of ‘spooky action at a distance’ and upholds the intuitive notion that cause and effect are spatially and temporally connected. The enduring appeal of local realism stems from its alignment with everyday experience and its successful application in describing a vast range of physical phenomena, making it a natural starting point for understanding the seemingly bizarre predictions of quantum mechanics.

Local Hidden Variable (LHV) models represent a compelling attempt to reconcile the seemingly probabilistic nature of quantum mechanics with the deterministic principles of classical physics. These models propose that quantum phenomena aren’t inherently random, but rather governed by additional, unobserved variables – the ‘hidden variables’ – that fully determine a particle’s properties and behavior. Essentially, LHV theories suggest that what appears as quantum uncertainty is merely a reflection of our incomplete knowledge; if these hidden variables were known, predictions about particle outcomes could, in principle, be made with certainty. The framework aims to provide a more complete description of reality by supplementing the standard quantum mechanical description with these underlying determinants, effectively reinstating a classical worldview where particle behavior isn’t truly probabilistic but instead pre-determined by its initial conditions and local environment. However, it’s crucial to note that such models have faced significant challenges, particularly from experimental tests of Bell’s inequalities, which suggest that local realism – and therefore, simple LHV models – may not fully capture the intricacies of the quantum world.

Local Hidden Variable (LHV) models are built upon the principle of locality, a cornerstone of classical physics asserting that an object is directly influenced only by its immediate surroundings. This translates to a strict requirement within these models that any description of a physical state must be constructed solely from measurements performed locally – that is, without instantaneous signals traveling between distant points. The assumption precludes “non-local” interactions, effectively rejecting the possibility of faster-than-light communication or influence. Consequently, LHV models attempt to reproduce quantum mechanical predictions by proposing that seemingly random quantum events are, in reality, determined by these pre-existing, locally accessible ‘hidden’ variables, restoring a deterministic and spatially-constrained understanding of the universe. The success or failure of these models hinges on whether this strictly local framework can adequately account for the observed correlations in quantum systems, particularly those that appear to violate the constraints of classical information transfer.

Local hidden-variable models necessitate that the physics governing these variables is independent of the quantum state and particle number, implying a time- and state-independent velocity field dictates their evolution and determines the distribution of hidden variables over time.

Mapping the Hidden Dynamics: The Velocity Field Unveiled

A complete Local Hidden Variable (LHV) model requires a specification of the temporal evolution of its hidden variables. This evolution is mathematically described by a ‘velocity field’, which maps each point in the hidden variable space to a corresponding rate of change in that variable. The velocity field, often represented as a function $v(x,t)$ where $x$ denotes the hidden variable state and $t$ represents time, dictates how the probability distribution of hidden variables changes over time. Defining this field is crucial because it determines the predictive power of the LHV model and allows for calculations of expected measurement outcomes as a function of time and initial conditions.

The Continuity Equation, a fundamental principle governing the velocity field in LHV models, mathematically expresses the conservation of probability as the hidden variables evolve. This equation, generally represented as $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0$, states that the rate of change of probability density ($\rho$) at a given point is equal to the negative divergence of the probability flux ($\rho v$), where $v$ represents the velocity field. Effectively, this ensures that while the distribution of hidden variables may change over time, the total probability within the system remains constant; probability is neither created nor destroyed, only redistributed according to the dynamics defined by the velocity field. This conservation is a necessary condition for a physically plausible LHV model.

The accuracy of a Latent Hidden Variable (LHV) model is fundamentally contingent on the physical plausibility of its dynamics. Specifically, the evolution of the hidden variables, as defined by the velocity field, must align with established physical principles governing the system being modeled. For systems approximating non-interacting particles, the dynamics should reduce to those expected from free particle behavior, demonstrating consistency with known physical limits. Deviations from these expected dynamics indicate a flawed model, potentially leading to inaccurate predictions or interpretations of the underlying system. Therefore, validating the velocity field against the principles of physical interaction is a critical step in ensuring the reliability of any LHV model.

Faithful implementation of unitary transformations on a hidden-variable space necessitates a dimensionality that scales with the number of qubits, as the Bell local hidden-variable model can only accommodate dynamics for a single qubit, while larger hidden-variable spaces are required to represent more complex quantum systems.

The Limits of Determinism: A No-Go Theorem Emerges

The introduction of interactions, such as the Heisenberg interaction which describes spin interactions in materials, fundamentally alters the dynamics of a system beyond simple free particle evolution. This necessitates the inclusion of hidden variables to account for correlations between particles that are not explained by quantum mechanics alone. Specifically, these hidden variables must define a state for each particle, and their evolution under interaction must reproduce the observed statistical correlations. The complexity arises because these correlations are non-trivial; they cannot be expressed as a simple tensor product of individual particle states, meaning the hidden variables are not independent and require a correlated evolution under the interaction. This correlation is crucial for maintaining consistency with experimental results and defining a complete description of the system’s dynamics.

The dynamics governing hidden variables within Local Hidden Variable (LHV) models are not arbitrary; they must adhere to established physical laws to remain consistent with observed outcomes. Specifically, the transformation of these hidden variables under interaction – dictated by a group of unitaries, $G$ – must reproduce the probabilities predicted by quantum mechanics. This necessitates a self-consistent mapping between the hidden-variable space, $\Lambda$, and the operational measurements performed on the physical system. Any proposed dynamic for the hidden variables that violates these constraints – leading to predictions differing from experimentally verified quantum results – renders the LHV model untenable. Therefore, the allowed transformations of hidden variables are fundamentally limited by the need to reconcile the underlying hidden-variable structure with observed physical reality.

A ‘No-Go Theorem’ establishes limitations for Local Hidden Variable (LHV) models attempting to reconcile quantum mechanics with classical determinism. This theorem proves that no LHV model can simultaneously uphold locality-the principle that distant events cannot instantaneously influence each other-realism-the existence of definite properties independent of measurement-and consistency of dynamics-the preservation of physical laws under transformations. The theorem mathematically formalizes this incompatibility through a Dimensionality Constraint: $dimG ≤ dimΛ(dimΛ+1)/2$, where $dimG$ represents the dimensionality of the group of unitaries describing the system’s evolution and $dimΛ$ is the dimensionality of the hidden-variable space. This inequality dictates that the complexity of the system’s dynamics is bounded by the size of the hidden-variable space required to explain its behavior, effectively limiting the potential dimensionality of the hidden-variable space itself.

The dimensionality of the group of unitaries, $G$, which mathematically describes the possible transformations of the system’s state, is fundamentally constrained by the dimensionality of the hidden-variable space, $\Lambda$. Specifically, the ‘No-Go Theorem’ establishes that the inequality $dimG ≤ dimΛ(dimΛ+1)/2$ must hold. This means that as the complexity of the hidden-variable space ($\Lambda$) increases, the allowable complexity of the unitary transformations ($G$) is also limited; a higher-dimensional hidden-variable space does not permit an arbitrarily complex group of transformations to be consistently defined while adhering to the principles of locality and realism. This relationship highlights a direct trade-off between the degrees of freedom available in describing the hidden variables and the permissible dynamics of the system.

Quantum states can be visualized as points within a disc, with separable and local states forming nested convex sets representing the range of correlations achievable under unitary evolution.

Beyond Local Realism: The Implications for Our Understanding of Reality

The bedrock of classical physics rests upon the principles of local realism – the idea that objects possess definite properties independent of observation, and that influences cannot travel faster than light. However, the No-Go Theorem fundamentally challenges this worldview, revealing an inherent incompatibility between local realism and the predictions of quantum mechanics. This theorem doesn’t simply suggest a flaw in our current understanding; it asserts that either locality – the principle that spatially separated events cannot instantaneously affect one another – or realism – the assumption that physical properties exist independently of measurement – must be relinquished to fully account for observed quantum phenomena. Essentially, the theorem demonstrates that the universe doesn’t operate according to the intuitive principles governing classical physics, forcing a re-evaluation of our most fundamental assumptions about reality.

The No-Go Theorem rigorously establishes the inadequacy of Local Hidden Variable (LHV) models – those attempting to explain quantum mechanics through predetermined properties and local interactions – when confronted with the full spectrum of quantum behaviors. These models, built on the premise of ‘Local States’, posit that each particle carries definite, albeit unknown, properties, and that measurement outcomes are determined solely by these local properties. However, the theorem demonstrates that no LHV model, regardless of the complexity of its hidden variables, can simultaneously reproduce all the statistical predictions of quantum mechanics, particularly those involving entangled particles. This isn’t merely a mathematical curiosity; it suggests that either the principle of locality or the principle of realism must be abandoned to fully account for the observed quantum world.

Investigations into Local Hidden Variable (LHV) models reveal a critical constraint on their ability to mimic quantum mechanics as system complexity increases. While LHV theories attempt to explain quantum correlations through pre-existing, local properties, the number of particles necessary to demonstrate a clear violation of these models-and thus, evidence for genuinely quantum behavior-doesn’t increase linearly with the complexity of the hidden-variable space. Instead, research indicates a logarithmic relationship: as the dimensionality of the space describing a particle’s hidden variables grows, the number of particles needed to expose the limitations of LHV dynamics increases at a decreasing rate. This logarithmic scaling suggests a fundamental limit to the scalability of LHV models, reinforcing the departure from classical realism.

The persistent challenge to local realism, underscored by recent theoretical work, doesn’t merely refine quantum mechanics-it compels a re-evaluation of fundamental assumptions about the nature of reality. The inability of Local Hidden Variable (LHV) models to fully account for observed quantum behavior suggests that either the principle of locality-the idea that an object is only directly influenced by its immediate surroundings-or realism-the belief that objects have definite properties independent of observation-must be relinquished. This isn’t simply a mathematical curiosity; it indicates that the universe may not operate according to the intuitive principles governing classical physics. Consequently, investigations into the foundations of quantum mechanics are expanding beyond the realm of purely predictive theory and are now actively probing the very structure of existence, forcing physicists and philosophers to reconsider long-held beliefs about causality, determinism, and the relationship between observer and observed.

The exploration into local hidden-variable models, as detailed in the study, isn’t merely a mathematical exercise; it’s an attempt to dismantle established assumptions about reality. One considers the implications if the apparent failures of these models aren’t limitations, but rather indications of a deeper, more complex structure. As John Bell once stated, “No phenomenon is a statement, but a test of a statement.” This sentiment perfectly encapsulates the research; the investigation doesn’t seek to prove the existence of local realism, but to rigorously test its boundaries. The work, by probing the limits of LHV models, suggests that quantum dynamics may necessitate abandoning classical intuitions about locality and independence, revealing a universe where interconnectedness reigns supreme.

Where Do We Go From Here?

The persistent failure to fully accommodate quantum dynamics within local hidden-variable models isn’t surprising, not really. It merely confirms a suspicion: reality is open source – the code is there, but the debugger isn’t cooperating. This work, by tightening the constraints on those models, doesn’t offer a solution, but it does refine the problem. Future investigations must move beyond simply seeking a local realistic description. The focus needs to shift to understanding why such a description consistently fails, especially as systems become more complex and entangled.

A critical limitation lies in the assumptions made about the nature of hidden variables themselves. Are these variables truly local, or is there some subtle, yet fundamental, non-locality embedded within their definition? Further research should explore more exotic forms of hidden variables, perhaps those incorporating elements of retrocausality or non-standard spacetime geometries. It’s a long shot, certainly, but clinging to intuition rarely advances understanding.

Ultimately, the question isn’t whether local realism can be salvaged, but what its limitations reveal. The continuing inability to reconcile quantum mechanics with classical intuition suggests a deeper, more fundamental structure governing reality, one that we’ve only glimpsed through the cracks in our current models. The code is there, waiting to be fully read, and the challenge now is to build the tools necessary to decipher it.

Original article: https://arxiv.org/pdf/2512.16682.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Unveiling Classical Foundations: The Illusion of Local Realism

Mapping the Hidden Dynamics: The Velocity Field Unveiled

The Limits of Determinism: A No-Go Theorem Emerges

Beyond Local Realism: The Implications for Our Understanding of Reality

Where Do We Go From Here?

See also: