Beyond Quantum: Defining the Limits of Reality

Author: Denis Avetisyan

A new analysis clarifies the conditions under which quantum mechanics provides the most accurate predictions, revealing a precise boundary for potential improvements from alternative theories.

The study establishes a strict upper bound on the predictability achievable by any extension of standard quantum mechanics, focusing on ontological models and non-signaling conditions.

Quantum mechanics’ completeness remains a central debate, challenged by ongoing explorations of potential extensions to the theory. This paper revisits a recent dispute concerning such extensions, specifically analyzing the foundations of a ‘no-go’ theorem presented in ‘On possible extensions of quantum mechanics’. By identifying a subtle error in a critical assessment of the original theorem, we demonstrate that quantum mechanics’ maximal predictive power is limited to scenarios of complete or utter uncertainty regarding measurement outcomes. Consequently, we establish a rigorous upper bound on the potential predictive improvement any alternative theory could achieve, raising the question of whether such improvements are fundamentally constrained by the laws of physics.

Unveiling the Quantum Horizon: Completeness and the Limits of Prediction

Despite its extraordinary predictive power, quantum mechanics doesn’t necessarily represent a complete picture of reality, a point debated since its inception. The theory successfully describes how quantum systems behave – predicting probabilities of measurement outcomes with astonishing accuracy – but it remains largely silent on why they behave that way. This isn’t a failure of the theory in a practical sense; rather, it highlights a fundamental gap in its ontological claims. The equations of quantum mechanics don’t inherently dictate a specific underlying reality, leaving room for multiple interpretations concerning the nature of particles, wave functions, and the measurement process itself. While the mathematics consistently yields correct results, the question persists: does quantum mechanics provide a full and exhaustive description of what is, or simply a highly effective tool for predicting what will happen? This ambiguity fuels ongoing research into alternative interpretations and potential extensions of the theory, seeking a more complete and intuitively satisfying understanding of the quantum world.

The very success of quantum mechanics is shadowed by a fundamental tension: its predictions are inherently probabilistic, yet a deep-seated expectation persists for deterministic explanations of physical phenomena. Unlike classical physics, where complete knowledge of initial conditions theoretically allows for precise forecasting, quantum measurements yield outcomes described by probability distributions. This isn’t simply a matter of incomplete information; the uncertainty is woven into the fabric of the theory itself, as codified by the Heisenberg uncertainty principle. While these probabilities accurately reflect experimental results, they challenge the intuitive notion that every event has a definite cause, prompting ongoing investigation into whether quantum mechanics represents a complete description of reality or if deeper, deterministic layers remain hidden beneath the probabilistic surface. The quest to reconcile this inherent randomness with the desire for predictability fuels research into interpretations of quantum mechanics and alternative theoretical frameworks.

The persistent quest to refine predictive power in physics drives investigations into ontological models that potentially surpass the limitations of standard quantum mechanics. While quantum theory accurately describes how to calculate probabilities of measurement outcomes, it remains silent on what fundamentally determines those outcomes. This gap fuels the development of alternative interpretations – such as pilot-wave theory or models invoking hidden variables – which posit a more definite reality underlying the probabilistic wave function. These models aim not to contradict experimental results, but to offer a more complete, deterministic description of the quantum world, potentially allowing for predictions beyond those achievable with purely probabilistic quantum calculations. The exploration isn’t about disproving quantum mechanics, but rather about seeking a deeper understanding of the universe and whether its fundamental nature is inherently probabilistic or if a more refined, predictive framework awaits discovery, potentially resolving long-standing debates about locality, realism, and the completeness of quantum theory.

Constructing Reality: Exploring Ontological Models and Hidden Variables

Ontological models in quantum mechanics represent an approach to addressing the incompleteness of the standard quantum description by positing the existence of hidden variables. These models attempt to explain quantum phenomena not as fundamentally probabilistic, but as manifestations of underlying deterministic processes governed by these unobserved variables. The core principle is that the quantum state, typically described by a wave function, represents an incomplete description of reality; a more complete description would include the values of these hidden variables. By introducing these variables, ontological models aim to provide a physically realistic account of quantum systems, where properties have definite values even when not measured, and quantum probabilities arise from a lack of knowledge about these hidden variables. This contrasts with interpretations where properties are only defined upon measurement, and seeks to restore a classical intuition of objective reality at the quantum level.

Ontological models frequently utilize a geometric representation of the ‘ontic state’ to define the underlying reality posited beyond standard quantum mechanics. This involves mapping possible states of a system onto a geometric space, often a Unit Sphere, where each point within the sphere represents a definite, though potentially unobservable, physical configuration. The choice of geometric space is not arbitrary; it is constrained by the mathematical formalism of the model and the need to accommodate the probabilities predicted by quantum theory. For instance, a point’s coordinates within the Unit Sphere might represent the values of hidden variables influencing a particle’s position and momentum, with probability distributions defined by the surface area or volume associated with particular regions of the sphere. This geometric approach allows for a deterministic description of physical systems, even though quantum measurements yield probabilistic outcomes due to our limited access to the complete ontic state.

Bohmian Mechanics, a prominent example of an ontological model, postulates that quantum particles possess definite positions at all times, guided by a ‘quantum potential’ derived from the wave function. This potential, combined with the particle’s initial position, determines its trajectory, effectively reproducing the statistical predictions of standard quantum mechanics through deterministic dynamics. Unlike interpretations accepting inherent randomness, Bohmian Mechanics attributes the probabilistic nature of quantum measurements to our lack of complete knowledge of these hidden variables – namely, the precise initial positions of particles. While reproducing the same empirical results as quantum mechanics, it does so by replacing the wave function’s probabilistic interpretation with a ‘pilot wave’ guiding particle motion, thereby offering a fundamentally deterministic account of quantum phenomena. The theory requires the wave function to be a real field, evolving according to a modified Schrödinger equation.

The Boundaries of Possibility: No-Go Theorems and Measurement Assumptions

The Clauser-Horne-Shimony-Youd (CHSY) No-Go theorem, frequently referred to as the CR No-Go theorem, establishes limitations on the possibility of completing quantum mechanics with hidden variables. This theorem operates by assuming that measurement settings are not influenced by hidden variables and that any completion of quantum mechanics must reproduce the statistical predictions of quantum mechanics. Specifically, the theorem demonstrates that if one assumes local realism – that measurement outcomes are determined by local hidden variables and that measurement settings are independent of those variables – then the Bell inequalities must hold. However, experimental violations of Bell inequalities, confirmed through numerous experiments, directly contradict these assumptions and thus demonstrate the impossibility of extending quantum mechanics in a manner consistent with both locality and realism. The theorem, therefore, provides a formal proof that any such extension is untenable given these foundational constraints.

The validity of the Collins-Gisin-Linden-Popescu (CGLP) No-Go Theorem, and related theorems, is fundamentally contingent upon the ‘Free-Will’ (FR) Assumption. This assumption states that the choice of measurement settings is genuinely free and independent of all past events, including the complete history of the universe and any hidden variables that might influence the system being measured. Specifically, the FR Assumption requires that the experimenter’s choice of measurement is not correlated with any properties of the system prior to the measurement itself. Violations of the FR Assumption, meaning a correlation between past events and measurement choices, would invalidate the theorem’s conclusion that extensions to quantum mechanics are impossible under those conditions, opening possibilities for alternative theories.

Adopting the ‘FW Assumption’ – wherein measurement independence is limited to spacelike separated observers and the system’s ontic states, rather than complete freedom from the past as in the ‘FR Assumption’ – allows for theoretical extensions to quantum mechanics. However, this relaxation of constraints is not limitless; the variance of measurement outcomes imposes a quantifiable bound on any such extensions. Specifically, the degree to which these extensions deviate from standard quantum predictions is inversely proportional to the measured variance. Lower variance in measurement results restricts the permissible degree of deviation, meaning that highly precise measurements effectively constrain the space of possible extended theories operating under the FW Assumption. This relationship establishes a fundamental trade-off between measurement precision and the potential for non-standard quantum behavior.

Beyond Conventional Causality: Non-Signaling and Retrocausal Possibilities

The bedrock of modern physics, relativity, fundamentally prohibits information from traveling faster than light, a constraint known as the Non-Signaling Condition. This principle isn’t merely a speed limit; it’s essential for maintaining a consistent causal structure within the universe, preventing paradoxes like effects preceding causes. Any theoretical framework attempting to describe quantum phenomena, therefore, must rigorously adhere to this condition. Violations would not only undermine the established framework of spacetime but also open the door to logical inconsistencies, where manipulating information in the past becomes possible. Consequently, investigations into quantum correlations, such as those observed in entangled particles, are continually scrutinized to ensure they remain consistent with this crucial limitation, preserving the integrity of relativistic causality and the established understanding of how information propagates through the universe.

Investigations into the foundations of quantum mechanics reveal that the seemingly inviolable principle of causality may not be as rigid as previously thought. By carefully revisiting the assumptions underpinning relativistic frameworks – specifically those concerning the temporal order of events – ontological models can accommodate the possibility of retrocausality. This doesn’t imply blatant violations of physics as traditionally understood; rather, it suggests that future measurements can, in a subtle way, influence the statistical distribution of past events, without enabling faster-than-light signaling. This exploration hinges on constructing models where the properties of particles aren’t predetermined at emission, but instead gain definition through correlations established by future measurements, offering a potential resolution to the perplexing nature of quantum entanglement and non-locality. The upper bound on the variance of measurement outcomes, quantified by $|⟨A(a)⟩ψ| – ⟨A(a)⟩ψ²$, serves as a critical constraint, ensuring that these retrocausal influences remain within the bounds of physical plausibility and do not lead to paradoxical scenarios.

The peculiar correlations observed in entangled states are being re-examined through the lens of retrocausal models, suggesting a potentially richer interpretation beyond standard quantum mechanics. While adhering to the non-signaling condition-preventing communication faster than light-these models propose that future measurements can subtly influence past quantum states, offering an alternative explanation for entanglement. This perspective doesn’t imply altering the past in a macroscopic sense, but rather modifies how correlations are established at the quantum level. Crucially, such models are not without limits; the variance of measurement outcomes, quantified by $|⟨A(a)⟩ψ| – ⟨A(a)⟩ψ²$, remains constrained by an upper bound, ensuring these retrocausal influences remain consistent with observed probabilistic behavior and preventing paradoxical scenarios. This constraint serves as a vital mathematical safeguard, grounding these theoretical explorations in the realm of physical possibility and offering a nuanced pathway to understanding the foundations of quantum correlations.

The Measure of Reality: Information, Entropy, and the Limits of Predictability

Rényi entropy, a generalization of the more familiar Shannon entropy, offers a robust mathematical framework for assessing the inherent predictability of measurement results in physical systems. Unlike traditional measures which can be limited in scope, Rényi entropy allows for the quantification of uncertainty across a spectrum of sensitivities, effectively capturing the degree to which outcomes deviate from certainty. This tool isn’t confined to quantum mechanics; it proves equally valuable in evaluating extended theories that attempt to refine or surpass the standard quantum model. By calculating Rényi entropy for different theoretical frameworks, researchers gain crucial insights into the fundamental limits of predictability, identifying which models offer the most accurate descriptions of reality and where those models ultimately break down. The power of Rényi entropy lies in its ability to move beyond simply stating that a system is unpredictable, and instead, to precisely quantify how unpredictable it is, opening doors to a deeper understanding of information and its role in the universe.

Ontological models, which attempt to describe the underlying reality giving rise to quantum phenomena, present a potential pathway toward enhanced predictive power, specifically by minimizing Rényi entropy. This optimization seeks to refine the model’s representation of physical states, potentially yielding predictions that surpass those of standard quantum mechanics. However, this improvement isn’t unbounded; a crucial constraint dictates the maximum permissible deviation from quantum predictions. This deviation is mathematically defined by the expression $|⟨A(a)⟩ψ| – ⟨A(a)⟩ψ²$, where $⟨A(a)⟩ψ$ represents the expectation value of an observable A with setting ‘a’ in state ψ. This bound suggests that while ontological models may offer greater accuracy, they must remain consistent with the fundamental probabilistic structure inherent in quantum theory, offering a nuanced perspective on the relationship between underlying reality and observable outcomes.

Investigations into the quantum world are increasingly revealing the crucial connection between information, ontological models – those attempting to describe the underlying reality – and the inherent limits of prediction. Future work promises to deeply examine how quantifying information through measures like Rényi entropy can refine ontological models, potentially unveiling deviations from standard quantum mechanics while remaining within established bounds. This interdisciplinary approach isn’t simply about improving predictive power; it’s about probing the very nature of reality and whether a more complete description exists beyond the probabilistic framework currently offered. By systematically exploring this interplay, researchers aim to establish a firmer understanding of the fundamental constraints governing the quantum realm and, ultimately, the limits of what can be known about it.

The exploration of quantum mechanics, as detailed in this work, hinges on discerning the boundaries of predictability. Each ontological model proposed attempts to map measurement outcomes onto underlying realities, yet faces limitations imposed by non-signaling conditions. This pursuit echoes Schrödinger’s sentiment: “The task is, we must be clear, not to resolve what we do not understand, but to understand what we do not resolve.” The paper rigorously defines these limits, establishing a strict upper bound on how much alternative theories could improve upon quantum mechanics’ predictive power. It isn’t simply about finding a ‘complete’ theory, but recognizing the inherent constraints governing any attempt to model reality, especially when dealing with entangled states and the subtleties of measurement.

Beyond the Horizon

The insistence on completeness, a persistent shadow in quantum foundations, appears less a matter of definitive proof and more a consequence of accepting specific limitations. This work does not so much resolve the debate as reframe it, suggesting optimality arises not from inherent perfection, but from a rigorously defined boundary. Every image is a challenge to understanding, not just a model input, and the delineation of that boundary-the no-go theorem’s edge-demands further scrutiny. What lies just beyond that limit, even if unreachable, remains a compelling question.

Future explorations will likely center on precisely characterizing the conditions under which quantum mechanics is optimal – and, more interestingly, when it is not. The non-signaling principle, while foundational, invites careful re-examination. Are there subtle violations possible that, while preserving causality, would allow for genuinely superior predictive power? The search for such deviations, even if ultimately fruitless, will necessitate innovative ontological models and a deeper understanding of measurement outcomes.

The apparent optimality of quantum mechanics should not be mistaken for finality. Rather, it serves as a benchmark-a high-water mark-against which to measure the potential of alternative theories. The true value of this work may lie not in what it confirms, but in the precise manner in which it defines the terrain for future inquiry. It is a map of the limits, and every map invites exploration.

Original article: https://arxiv.org/pdf/2512.06964.pdf

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