Beyond Certainty: When Quantum Measurements Align—and Diverge

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Author: Denis Avetisyan

A new analysis explores the fundamental limits of agreement between observers making measurements on quantum systems, revealing that disagreement isn’t just possible, but governed by predictable bounds.

The scenario illustrates a distributed system involving two laboratories.
The scenario illustrates a distributed system involving two laboratories.

This review details the Quantum Agreement Theorem and its implications for understanding intersubjectivity and epistemic states in quantum mechanics.

Establishing shared knowledge is fundamental, yet quantum mechanics seemingly permits discrepancies in agents’ probabilistic beliefs about the same system. This paper presents ‘The Quantum Agreement Theorem’, rigorously defining when such differing beliefs can persist, building upon classical epistemic frameworks to explore a hierarchy of ‘common certainty’ operators. We demonstrate that while recording measurement outcomes restores classical agreement, quantum mechanics allows for bounded, non-classical disagreement between agents-a phenomenon we term common certainty of disagreement. Does this framework offer a pathway toward resolving long-standing questions regarding intersubjectivity and the foundations of quantum measurement?

The Paradox of Shared Reality

Quantum mechanics, despite its extraordinary predictive power, fundamentally clashes with classical notions of objectivity and shared reality. Historically, science operated under the assumption that properties of objects exist independently of observation, allowing for universal agreement on their values. However, quantum theory posits that certain properties aren’t definite until measured, and the very act of measurement influences the system. This isn’t merely a limitation of detection; it suggests that reality, at its most fundamental level, isn’t a pre-existing state “out there” waiting to be discovered, but is instead shaped by the interaction between the observer and the observed. Consequently, establishing agreement on quantum properties becomes problematic, as different observers, or even the same observer making repeated measurements, may obtain different outcomes, not due to faulty instruments, but due to the inherent probabilistic nature of quantum events and the observer’s role in defining them. This challenges the bedrock assumption of a universally accessible and objective reality upon which classical science is built.

Quantum mechanics diverges sharply from classical physics by positing that measurement outcomes are not pre-existing properties of a system, but rather are defined through the interaction between the system and the observer. This isn’t simply a matter of observational limitations; the observer’s state – encompassing the measuring device and even the broader experimental setup – fundamentally influences the result. Consequently, a particle doesn’t possess a definite value for a property like position or momentum until measured; the act of measurement creates the observed value relative to the observer’s specific frame of reference. This relational quality means that different observers, even with identical setups, might obtain different outcomes, not due to error, but because the quantum state is only realized through the observation itself. The implications are profound, suggesting that objectivity, as traditionally understood, is not absolute, and reality is, in a very real sense, observer-dependent.

The foundations of a shared reality are subtly undermined by quantum mechanics, prompting a re-evaluation of what it means for observers to agree on a measurement. Classical physics assumes an objective truth exists independently of observation, allowing for agreement based on comparing notes about this pre-existing reality. However, quantum theory posits that properties aren’t definite until measured, and the act of measurement inherently depends on the observer’s state. This isn’t merely a practical limitation, but a fundamental challenge: if reality is, in some sense, created by the observation itself, then agreement isn’t about matching descriptions of a common external world, but about the correlation of subjective experiences. Determining whether two observers “agree” becomes less about identifying a shared object and more about establishing a consistent relationship between their individual, observer-dependent realities, demanding a novel framework for understanding consensus in a universe where objectivity is not guaranteed.

Traditional protocols for establishing agreement, such as comparing notes or seeking consensus, rely on the assumption of pre-existing, objective properties independent of observation. However, quantum mechanics fundamentally undermines this premise. Due to the inherent uncertainties governed by the Heisenberg uncertainty principle and the observer-dependent nature of measurement – where the act of observing influences the observed – attempts to verify shared knowledge using classical methods become problematic. For example, two observers measuring a quantum system may obtain different outcomes, not due to error, but because the measurement itself forces the system into a definite state relative to each observer. This isn’t simply a matter of imperfect information; it challenges the very definition of “agreement” when the properties being measured don’t possess a pre-defined value until observed, rendering classical consensus mechanisms ineffective in the quantum realm. Consequently, new frameworks are required to establish meaningful agreement, acknowledging and accounting for the fundamental role of observation and the probabilistic nature of quantum reality.

Formalizing the Boundaries of Certainty

Common certainty represents a higher-order epistemic state beyond simple belief. It requires not only that an agent believes a proposition to be true, but also that the agent believes that all other relevant agents also believe the proposition to be true. This isn’t merely a belief about another’s belief; it’s a certainty that the belief is shared. Formally, if agent A believes proposition $P$, and A believes that every other agent B also believes $P$, and A believes that every agent knows that every agent knows $P$, and so on, then A possesses common certainty regarding $P$. This recursive condition distinguishes common certainty from merely shared knowledge, and forms a foundational requirement for coordinating action and achieving agreement in multi-agent systems.

Game Theory provides a formalized system for representing and analyzing agents’ beliefs about each other’s knowledge. Specifically, it utilizes concepts like common knowledge – where a fact is known to all, everyone knows that everyone knows it, and so on – and models these epistemic states using mathematical structures. These structures often involve probability distributions over possible states of the world, allowing for reasoning about incomplete information and uncertainty. Formal models, such as those built on Bayesian networks or decision trees, enable the prediction of rational behavior given assumptions about agents’ knowledge and preferences. The framework allows for the rigorous investigation of how shared knowledge influences strategic interactions and the emergence of agreement or disagreement between rational actors, moving beyond intuitive understandings of shared beliefs.

The Classical Agreement Theorem, originating in distributed computing, specifies conditions under which a group of agents can reliably reach a consensus on a single value, given they each have initial private inputs. These conditions typically involve a majority of agents agreeing and a mechanism to prevent conflicting decisions. However, the theorem’s assumptions – primarily concerning deterministic communication and shared knowledge of the system’s state – are fundamentally violated in quantum systems. Quantum communication is probabilistic, and the act of measurement introduces inherent uncertainty and alters the system’s state, precluding the deterministic assumptions necessary for the Classical Agreement Theorem to hold. Consequently, directly applying the theorem to scenarios involving quantum information exchange or shared quantum states yields inaccurate or inapplicable results, necessitating alternative frameworks for analyzing agreement in quantum contexts.

The framework utilizes the Born-Lüders rule to model how quantum states are updated following a measurement by an observer. This allows for a dynamic assessment of certainty, as the state reflects new information gained through observation. Analysis of this state update mechanism demonstrates that disagreement between observers is not unbounded; the maximum permissible disagreement is quantified as a value between 0 and 1. This bound arises from the probabilistic nature of quantum measurement and the way the Born rule dictates state collapse, limiting the divergence of observers’ knowledge even with differing measurement outcomes.

Defining the Limits of Disagreement

Zero-One Disagreement defines a scenario in which two agents possess entirely contradictory beliefs regarding an event; one agent assigns a probability of 1 (certainty) while the other assigns a probability of 0 (impossibility). This represents the maximum possible deviation in probabilistic assessment between two observers. Such complete disagreement isn’t merely a large difference in opinion, but a fundamental incompatibility in their belief states regarding the outcome. This extreme case serves as a useful benchmark for understanding the limits of probabilistic divergence and for establishing the boundaries within which meaningful agreement, or at least a constrained form of disagreement, can exist. It’s important to note that while theoretically possible, the framework employing signed probabilities and the No-Signaling Principle actively constrains the degree to which agents can approach this absolute form of disagreement, especially within quantum systems.

The No-Signaling Principle, a core tenet of relativistic physics, dictates that no information can travel faster than the speed of light. This constraint has direct implications for the potential disagreements between observers regarding quantum measurements. Specifically, it prevents one observer’s measurement from instantaneously influencing the probabilities assigned by another observer to outcomes in a separate, spacelike-separated region. Consequently, the degree to which two observers can disagree about quantum states is fundamentally limited; any observed correlation must be consistent with information transfer occurring at or below the speed of light. Violations of this principle would imply superluminal communication, a scenario precluded by established physical laws and therefore restricting the scope of permissible disagreements in quantum mechanics.

To model constrained probabilities arising from limitations on disagreement, we utilize signed probabilities, a formalism where negative values indicate the impossibility of an outcome. Rigorous analysis demonstrates that the maximum deviation between the probability estimates of two observers, denoted as observers A and B, is bounded by the expression $2||ρA – ρB||1/cA$. Here, $ρA$ and $ρB$ represent the density matrices describing the states of observers A and B, respectively, $||ρA – ρB||1$ denotes the trace norm of the difference between these density matrices, and $cA$ is a constraint factor quantifying the degree of allowed disagreement. This bound establishes a quantifiable limit on the extent to which differing observers can assign probabilities to the same event, given the constraints imposed by the No-Signaling Principle.

By utilizing signed probabilities and the No-Signaling Principle, this framework establishes quantifiable limits on permissible deviations between probability estimates held by different observers in quantum scenarios. The maximum allowable discrepancy is mathematically defined as $2||ρA – ρB||1/cA$, where $ρA$ and $ρB$ represent the density matrices of the two observers and $cA$ is a constant related to the system’s coherence. This allows for a precise determination of whether observed disagreements are consistent with the fundamental constraints of quantum mechanics, effectively delineating the boundaries of valid agreement and identifying scenarios where observed discrepancies would necessitate a revision of underlying assumptions or the introduction of new physical principles.

Towards an Observer-Centric Understanding of Agreement

Certain interpretations of quantum mechanics, notably Relational Quantum Mechanics and QBism, fundamentally challenge the classical notion of an objective reality independent of observation. These frameworks posit that a quantum state isn’t a pre-existing property of a system, but rather a representation of an observer’s degree of belief about potential measurement outcomes. This isn’t merely an epistemological stance-a statement about knowledge-but an ontological one, concerning the very nature of being; reality, in these views, is observer-dependent. Consequently, different observers, engaging with a quantum system, may legitimately assign different states to it, not due to flawed measurement, but because the state is defined relative to their individual interactions and information. This perspective shifts the focus from seeking an observer-independent ‘true’ state to understanding how information is exchanged and updated between observers, paving the way for a more consistent account of quantum phenomena and the emergence of intersubjective agreement.

Contemporary interpretations of quantum mechanics, such as Relational Quantum Mechanics and QBism, depart from the classical notion of objective reality by positing that a system’s state isn’t a pre-existing property but emerges from the interaction between the system and an observer. This perspective fundamentally shifts the focus from intrinsic characteristics to relational ones; a particle doesn’t have a definite position until it is measured – or, more accurately, until an interaction establishes a definite relationship between the observer and the particle. The state, therefore, becomes a representation of an observer’s degree of belief or information gained through that interaction, rather than an inherent attribute of the system itself. This relational quality means different observers, engaging with the same system, may legitimately assign different states, not due to error, but because the state is always defined relative to the specific interaction and the observer’s perspective, necessitating a re-evaluation of how agreement and objectivity are understood in the quantum realm.

The ‘Friend Paradox’ arises within quantum mechanics when considering scenarios involving entangled observers, revealing a surprising disconnect between individual experiences and collective agreement. This thought experiment demonstrates that if two observers independently measure a quantum system in an entangled state, each can legitimately conclude the other observed a different outcome, even though their measurements are correlated. The paradox isn’t a contradiction, but rather a consequence of assigning quantum states to observers themselves, implying that observation isn’t a passive recording of pre-existing properties. Resolving this necessitates a refinement of how agreement is defined in quantum contexts, moving beyond simple correspondence of results to a more nuanced understanding of conditional probabilities and information exchange. This revised framework is crucial for reconciling individual observer experiences with the consistent, shared reality predicted by quantum theory, and offers a pathway to a more robust account of intersubjectivity where observation itself plays a fundamental role.

Quantum mechanics, when viewed through observer-centric interpretations, necessitates a re-evaluation of how agreement between different observers is established. Rather than assuming a pre-existing, objective reality, these frameworks posit that quantum states are defined by the interaction between a system and an observer. This perspective allows for a more consistent account of intersubjectivity, moving beyond simple correspondence of measurement outcomes. Recent work further demonstrates the robustness of this agreement, even with slight variations in observers’ information or perspectives; utilizing a parameter $ϵ$ to quantify these perturbations, it has been shown that the deviation between probability estimates can be rigorously bounded by $2||ρA – ρB||1/cB$, where $ρA$ and $ρB$ represent the density matrices of observers A and B, and $cB$ is a constant related to the system’s coherence. This mathematical constraint suggests that even with small differences in observers’ knowledge, a high degree of agreement in quantum measurements remains predictable and stable, solidifying the foundation for a nuanced understanding of shared quantum experiences.

The exploration of agreement and disagreement in quantum measurements, as detailed in the paper, necessitates a reduction to fundamental principles. It reveals that certainty, even in shared observation, isn’t absolute, but rather bounded by the inherent probabilistic nature of quantum mechanics. This aligns with the sentiment expressed by Richard Feynman: “The first principle is that you must not fool yourself – and you are the easiest person to fool.” The paper meticulously outlines these bounds, stripping away assumptions of complete consensus to reveal the underlying conditions under which agents can reliably agree, or, crucially, understand the limits of their potential disagreement. The work’s strength lies in its refusal to overcomplicate; it seeks not to impose agreement, but to precisely define its possibility, and the allowable space for divergence.

Further Refinements

The demonstration that quantum certainty need not equate to universal agreement, while logically sound, exposes a persistent tension. The paper establishes bounds on disagreement, but does not illuminate the pragmatic implications of such divergence. Future work must address whether, and under what circumstances, quantifiable disagreement constitutes a resource-or merely noise-in multi-agent quantum systems. The exploration of ‘quantum disagreement’ as a fundamental quantity, analogous to entanglement or coherence, remains largely uncharted territory.

A critical limitation resides in the idealized conditions assumed. The theorem operates on epistemic states, abstracting away from the messy realities of imperfect state knowledge and noisy communication channels. A natural progression involves relaxing these assumptions, investigating how practical constraints erode the bounds on agreement, and establishing the minimum level of shared information required to achieve reliable coordination. Unnecessary complexity in modeling such imperfections, however, is violence against attention.

Ultimately, the question is not merely whether agents can agree, but why they would seek to do so. The paper’s formalism provides a foundation for analyzing scenarios where strategic disagreement-the deliberate cultivation of divergent beliefs-might be advantageous. This line of inquiry, while speculative, hints at a deeper connection between quantum mechanics, game theory, and the very nature of information itself. Density of meaning is the new minimalism.

Original article: https://arxiv.org/pdf/2511.21258.pdf

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

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2025-11-28 05:10