Beyond Quantum: The Geometry of Reality

Author: Denis Avetisyan

New research reveals how discrete mathematical structures are reshaping our understanding of quantum phenomena and the very foundations of information processing.

The Peres-Mermin square-a construction leveraging two-qubit Pauli matrices-demonstrates the inherent contextuality arising from the orthogonality relations within Peres’s 24-ray scenario, revealing that quantum mechanics cannot be consistently described by non-contextual hidden variable theories.

This review explores the use of hypergraphs and causal models to characterize quantumness, contextuality, and entanglement.

While quantum theory demonstrably departs from classical probability, quantifying this ‘quantumness’ remains a central challenge in foundational research. This thesis, ‘Quantumness via Discrete Structures’, investigates the role of discrete mathematical structures-specifically graphs, hypergraphs, and directed acyclic graphs-in characterizing and operationalizing quantum phenomena. By applying these tools to contextuality, causality, and measurement incompatibility, we reveal deeper connections between quantumness and fundamental limitations on information processing. How might these discrete frameworks ultimately illuminate the boundary between the quantum and classical worlds, and guide the development of novel quantum technologies?

Beyond the Classical Horizon: Unveiling Deeper Correlations

Despite its extraordinary predictive power, standard quantum theory may represent an incomplete picture of correlation’s full potential. Investigations reveal that the quantum realm allows for correlations exceeding those permitted by classical physics, yet these aren’t necessarily encompassed within the established framework of quantum mechanics. Researchers are discovering correlations that, while experimentally verified, lack a straightforward description using conventional quantum states and measurements – suggesting the existence of a broader class of correlations beyond those typically considered. This implies that the informational relationships between quantum systems are richer and more complex than previously understood, potentially necessitating an expansion of the theoretical tools used to describe them and hinting at a deeper, underlying structure governing the connections between physical entities. The quantification of these ‘extra’ correlations, through the use of invariants, provides a pathway towards characterizing and potentially harnessing these novel phenomena.

Investigations into quantum systems increasingly reveal correlations that defy explanation through classical physics. These aren’t simply statistical coincidences; rather, they represent connections between events that are demonstrably stronger than any permitted by local realism – the intuitive notion that an object’s properties are predetermined and can only be influenced by its immediate surroundings. Experiments consistently demonstrate violations of Bell’s inequalities, a mathematical formulation of local realism, indicating that quantum systems exhibit entanglement – a phenomenon where particles become linked in such a way that they share the same fate, no matter how far apart they are. Such nonclassical correlations aren’t limited to isolated systems; they manifest in diverse scenarios, from the behavior of photons to the interactions of complex molecules, suggesting that these effects may be fundamental to the universe’s informational structure and potentially unlocking new avenues in quantum technologies. The persistent observation of these correlations compels a re-evaluation of the boundaries of classical understanding and points toward deeper, yet unexplored, aspects of reality.

The pursuit of quantifying nonclassical correlations isn’t merely an exercise in mathematical formalism; it represents a fundamental effort to map the boundaries of reality and its inherent informational content. Researchers are increasingly focused on identifying and measuring these correlations using mathematical tools called invariants – quantities that remain unchanged under certain transformations. These invariants offer a means to characterize the degree of nonclassicality, essentially providing a numerical value representing how far a system deviates from what classical physics predicts. This approach moves beyond simply observing strange phenomena to actively measuring the depth of their departure from classical expectations, with implications for fields ranging from quantum information theory to our foundational understanding of spacetime and the very structure of information itself. By rigorously defining these limits, scientists are gaining unprecedented insight into the informational scaffolding of the universe and potentially unlocking new avenues for technological innovation.

This protocol enables Alice to reliably transmit a message to Bob by encoding it in her measurement choice on an entangled state, sending the result through a classical noisy channel, and allowing Bob to decode the message via a subsequent quantum measurement and post-processing.

Mapping the Quantum Landscape: Hypergraphs and Contextuality

Contextuality, in quantum mechanics and related theories, refers to the dependence of measurement outcomes not solely on the measured property itself, but also on the set of compatible measurements performed alongside it. This deviates from classical physics, where properties are assumed to have definite values independent of the measurement process. Specifically, a measurement outcome for a given observable can vary depending on which other mutually compatible observables are measured simultaneously. This interdependence implies that assigning pre-defined values to observables independent of the measurement context is not possible, and any attempt to do so leads to contradictions with experimental results. The presence of contextuality is therefore considered a hallmark of nonclassicality, distinguishing quantum systems from those governed by classical principles where outcomes are predetermined by inherent properties.

Hypergraphs offer a mathematical formalism for representing the compatibility relations between measurement settings in quantum mechanics, formalized through the $JointMeasurabilityStructure$ (JMS). Unlike traditional graphs which represent pairwise relations, hypergraphs allow for the representation of relationships involving any number of measurement settings. The JMS is constructed by defining a hyperedge for each maximal set of compatible measurements; vertices represent individual measurement settings, and a hyperedge connects all settings that can be jointly measured without contradiction. This allows for a comprehensive mapping of which measurements can be performed together, and forms the basis for quantifying contextuality by analyzing the hypergraph’s structure and connectivity. The use of hypergraphs simplifies the analysis of complex measurement scenarios and provides a tool to determine the limits of classical predictability.

Contextuality, as a departure from classical physics, requires quantifiable metrics for rigorous analysis; WeightedMaxPredictability (WMP) serves as such an invariant. WMP assesses the maximum predictability of outcomes given all possible measurement settings, and its value is constrained by classical bounds. Specifically, a WMP value exceeding these classical limits indicates the presence of contextuality, demonstrating a violation of assumptions inherent in classical models. Researchers utilize WMP to not only detect contextuality but also to quantify its degree, providing a numerical measure of non-classical behavior and enabling comparative analysis of different quantum systems and measurement scenarios. This approach allows for a more precise characterization of non-classicality than traditional methods, offering a robust framework for exploring the foundations of quantum mechanics and its applications.

The depicted scenario, Γ5, illustrates contextuality within the KCBS framework as described in reference [36].

Defying Temporal Order: Indefinite Causal Structures

OperationalQuantumTheory, unlike classical frameworks, permits the construction of models where the temporal order of events is not predetermined. This leads to the phenomenon of IndefiniteCausalOrder, where the sequence in which events occur is fundamentally undefined until measurement. This is not simply a matter of incomplete knowledge; rather, the theory allows for consistent descriptions of physical processes where assigning a definite temporal order is logically impossible. Such scenarios are mathematically permissible within the framework and do not necessarily violate fundamental physical principles, but necessitate a departure from classical notions of cause and effect. The possibility of IndefiniteCausalOrder arises from the theory’s focus on operational definitions and the rejection of pre-existing assumptions about temporal structure.

Representing indefinite causal structures necessitates the use of tools beyond standard spacetime diagrams. Directed Acyclic Graphs (DAGs) provide a framework for modeling potential causal relationships where the temporal order between events is not definitively established. In a DAG, nodes represent events and directed edges signify potential causal influence, with the acyclic constraint preventing causal loops. These graphs do not impose a fixed temporal order; instead, they represent a set of possible causal relationships, each with an associated probability. The structure of the DAG, therefore, encodes the allowed causal influences, and probabilities are assigned to each possible ordering consistent with the graph. Analyzing these graphs allows for the computation of probabilities for different causal scenarios and the quantification of the degree of causal ambiguity present in the system, which is crucial for evaluating the implications of indefinite causal order.

Antinomicity, observed within indefinite causal structures, represents a deviation from classical causality where definite outcomes are not predetermined prior to measurement. This phenomenon manifests as the possibility of assigning conflicting causal orderings to events, challenging the assumption of a universally valid temporal structure. Specifically, this work demonstrates a quantifiable link between antinomicity and Bell nonlocality; the degree of antinomicity is directly correlated with the violation of Bell inequalities. This connection indicates that the ability to define a consistent causal order is not a prerequisite for physical reality, and that nonlocal correlations are fundamentally intertwined with the absence of a fixed temporal structure. The presence of antinomicity, therefore, provides an operational signature of the limitations of classical causality and its relationship to quantum nonlocality.

The adversarial feedback (AF) and beneficial weighting (BW) process exhibits a reciprocal causal relationship, where each party influences and is influenced by the others.

Beyond Classical Intuition: Kochen-Specker and the Nature of Reality

Kochen-Specker contextuality reveals a profound departure from classical intuition regarding the measurability of quantum properties. The theorem demonstrates that assigning definite values to all quantum observables – properties like spin or polarization – leads to inescapable logical contradictions. This isn’t a limitation of measurement technology, but a fundamental property of the quantum world itself. Consider a scenario where measuring the value of one observable intrinsically alters the possible outcomes of measuring another, not due to physical disturbance, but because the very definition of those properties depends on the measurement context. Mathematically, this is illustrated by constructing sets of incompatible measurements where assigning a definite value to each measurement would violate the law of non-contradiction – a statement cannot be both true and false simultaneously. This contextuality challenges the classical assumption that quantum properties have pre-existing, definite values independent of observation, suggesting instead that reality, at the quantum level, is inherently relational and defined by the act of measurement.

The implications of Kochen-Specker contextuality extend far beyond the realm of mathematical formalism, challenging deeply held assumptions about the nature of reality itself. Classical physics operates on the premise that properties of a system exist independently of measurement – a definite value is inherent, even if unknown. However, this contextuality demonstrates that, in the quantum world, the very act of measurement fundamentally influences the property being observed, meaning a quantum system doesn’t possess pre-defined values for all its properties simultaneously. This isn’t simply a limitation of our ability to know, but an intrinsic feature of the quantum world; the system’s properties are defined only within the context of the measurement being performed. Consequently, this suggests that reality at the quantum level isn’t composed of pre-existing, objective properties, but rather emerges from the relationships between the system and the observer, blurring the lines between observer and observed and demanding a re-evaluation of what it means for something to “exist” independently of measurement.

Specker’s principle reveals a deep tension between the intuitive classical worldview and the predictions of quantum mechanics, asserting that assigning definite values to all measurable properties of a quantum system simultaneously leads to logical inconsistency. This isn’t merely a philosophical point; demonstrable links connect this principle to the violation of Bell inequalities, a cornerstone of quantum nonlocality. Specifically, the impossibility of satisfying Specker’s constraints mirrors the impossibility of creating a local hidden variable theory that reproduces quantum correlations. Experiments verifying Bell inequality violations thus provide empirical evidence supporting Specker’s principle and, consequently, the fundamental incompatibility of classical realism with the observed behavior of quantum systems. The work suggests that quantum mechanics necessitates a rejection of the assumption that properties exist independently of measurement, challenging deeply held notions about the nature of reality itself.

The 18-ray contextuality scenario, denoted as Γ18 and detailed in Ref. [34], is utilized to explore contextual effects.

The Quantum Horizon: Future Applications and Beyond

Entanglement-assisted communication represents a paradigm shift in how information can be transmitted, leveraging the unique correlations of quantum entanglement to potentially overcome limitations inherent in classical systems. This approach doesn’t aim to replace existing communication methods, but rather to supplement them; by pre-sharing entangled quantum states – where two or more particles become linked and share the same fate, no matter how far apart – communication protocols can be designed to increase channel capacity or enhance security. Specifically, certain communication tasks, like sending information reliably through noisy channels, can be performed more efficiently with entanglement than is possible classically. While still largely theoretical and facing significant technological hurdles, research demonstrates that by carefully exploiting these quantum correlations, it may be possible to transmit information with greater robustness and potentially at higher rates, opening avenues for future advancements in secure data transfer and high-bandwidth communication networks.

The practical application of quantum concepts, such as entanglement-assisted communication, fundamentally depends on the ability to create and control qubits – the basic units of quantum information. This work demonstrates significant progress in QubitRealization, moving beyond theoretical models to physically implement quantum structures. Researchers successfully constructed and verified structures comprising up to four vertices, a crucial step towards building more complex quantum systems. This achievement relies on precise control over individual qubits and their interactions, paving the way for scalable quantum technologies. The realization of these multi-vertex structures confirms the feasibility of building the hardware necessary to harness nonclassical correlations for future applications, and offers a tangible foundation for further advancements in quantum information science.

The progression of quantum research increasingly suggests a future fundamentally reshaped by the practical application of nonclassical correlations. Beyond theoretical exploration, recent advancements in qubit realization – demonstrated through the successful construction of quantum structures with up to four vertices – are actively translating these principles into tangible possibilities. This isn’t simply about faster computation; the harnessing of entanglement and other quantum phenomena promises breakthroughs across diverse fields. Secure communication networks impervious to eavesdropping, ultra-sensitive sensors capable of detecting previously imperceptible signals, and materials with entirely novel properties are all within reach. The ability to manipulate and utilize these uniquely quantum effects represents a paradigm shift, moving beyond the limitations of classical physics and ushering in an era of transformative technologies poised to redefine industries and daily life.

The exploration of quantumness through discrete structures necessitates a dismantling of classical intuitions. This thesis, by focusing on hypergraphs and their capacity to model contextuality, actively tests the boundaries of established causal reasoning. One might recall Werner Heisenberg’s assertion: “Not only must one correct the action of the apparatus, but one must also correct the theory.” The work echoes this sentiment; it isn’t enough to simply apply existing mathematical tools. Instead, the very foundations of those tools – how they represent information and measurement – are subject to rigorous examination. By probing the limits of graph-based representations of quantum phenomena, this research demonstrates that true understanding arises from challenging and reconstructing the framework itself, a principle central to unlocking the mysteries of entanglement and Bell nonlocality.

What Lies Ahead?

The exploration of quantumness through the lens of discrete structures, while promising, inevitably reveals the limitations of any attempt to fully capture quantum phenomena with classical tools. The very act of representing contextuality, entanglement, or nonlocality via graphs and hypergraphs highlights what is necessarily lost in translation – the inherent dynamism and counter-intuitive nature of the quantum realm. Future work should not aim for perfect mimicry, but rather focus on identifying precisely where and how these discrete models break down, and what those failures reveal about the underlying quantum principles.

A particularly fertile avenue lies in moving beyond static representations. The current models largely treat quantum states as fixed entities projected onto a discrete structure. A more nuanced approach would involve incorporating temporal dynamics – exploring how these structures evolve under quantum operations, and whether the topology itself can serve as a resource for quantum computation. The question isn’t simply can we represent quantumness classically, but what can we learn by deliberately pushing the boundaries of that representation until it collapses?

Ultimately, the value of this approach may not be in constructing a “quantum simulator” based on graphs, but in revealing the fundamental inadequacies of our classical intuitions. True security, in this context, resides not in obscuring the underlying mechanisms, but in explicitly acknowledging their limitations – a transparency that allows for a more honest and fruitful exploration of the quantum world.

Original article: https://arxiv.org/pdf/2512.10063.pdf

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