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Phenomenological quantum mechanics: I. Phenomenology of quantum observables

This paper proposes a phenomenological approach to deriving quantum mechanics solely from sequential measurement data, demonstrating that such observations necessitate a non-classical "bi-trajectory" formalism rooted in Hilbert spaces rather than a standard uni-trajectory state description.

Original authors: Piotr Szańkowski, Davide Lonigro, Fattah Sakuldee, Łukasz Cywiński, Dariusz Chruściński

Published 2026-08-18

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CC BY 4.0

Original authors: Piotr Szańkowski, Davide Lonigro, Fattah Sakuldee, Łukasz Cywiński, Dariusz Chruściński

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum mechanics is the rulebook for how the smallest things in the universe behave, from the atoms that make up a table to the light that travels from a distant star. For nearly a century, scientists have used a specific mathematical language to describe these tiny particles, a language that relies on a concept called a "state." In this standard view, a particle exists in a specific condition until someone looks at it, at which point the act of measurement forces it to choose a single outcome. This framework works incredibly well for predicting what happens in a single experiment. However, it struggles when scientists try to describe what happens when they measure the same particle over and over again in a sequence. The standard rules require a sharp, somewhat arbitrary line between the time the particle is evolving on its own and the time it is being measured, a division that feels unnatural when trying to understand the full story of a quantum system's life.

A team of researchers has now proposed a different way to look at this problem, one that starts not with the complex math of the standard theory, but with the raw data of what actually happens in the lab. Instead of assuming the existence of a "state" or a specific mathematical structure, they asked a simple question: if we only knew the results of a series of measurements performed one after another, what kind of theory would we be forced to invent to explain them? By treating the experimental outcomes as the only truth, they deduced a new formalism that describes quantum systems without ever needing to draw that arbitrary line between measurement and non-measurement. Their work suggests that the reason our current theory feels so strange and disjointed is that it is built on an incomplete picture of reality.

The researchers began by imagining a scenario where an experimenter has a collection of devices capable of measuring different properties of a quantum system. They recorded the results of countless sequences of measurements, noting the specific outcomes and the exact times they occurred. In a classical world, if you measure a property like position or speed repeatedly, you are simply sampling a single, continuous path that the object has been tracing through time. If you skip a measurement in the middle of the sequence, the results before and after should still fit together perfectly, as if that missing measurement never happened. This is a fundamental rule of classical physics: the past influences the future, but the future cannot change the past, and the history of an object is a single, consistent story.

However, when the researchers analyzed the data from quantum systems, they found that this single-story picture completely breaks down. When they tried to skip a measurement in the middle of a sequence, the results did not match what would happen if that measurement had simply been omitted. The act of measuring, even if the device is set to be less precise or "coarse-grained," fundamentally alters the history of the system in a way that cannot be explained by a single path. This violation of consistency means that a quantum system does not follow a single trajectory through time. Instead, the system seems to explore multiple possibilities simultaneously, and the final result depends on how these possibilities interfere with one another.

This interference is the key to understanding the new picture. The researchers found that the results of these sequential measurements could be perfectly described by considering pairs of paths rather than just one. In their new framework, known as the "bi-trajectory" picture, the probability of any outcome is determined by the interaction between two distinct sequences of events. One sequence represents the path the system took, and the other represents a partner path that interferes with it. These pairs of paths are not just mathematical tricks; they are the fundamental building blocks required to explain why quantum systems behave the way they do. The researchers demonstrated that if you assume the system follows a single path, you cannot explain phenomena like the quantum Zeno effect, where frequent measurements seem to freeze a system in place, or the uncertainty relations that prevent certain pairs of properties from being known simultaneously.

The beauty of this new approach is that it removes the need for the "collapse" of a state, a concept in standard quantum mechanics that describes how a system jumps from many possibilities to one definite result upon measurement. In the bi-trajectory picture, there is no sudden jump and no need to decide when a measurement begins or ends. The system is always described by the complex interplay of these paired paths. The researchers showed that this description is not just a different way of writing the same equations, but a more fundamental way of understanding the physics. They proved that the standard rules of quantum mechanics can be derived from this new picture, but the new picture also handles complex scenarios involving multiple systems and sequential measurements much more naturally.

By starting with the phenomenology—the actual observed behavior of the system—the team was able to construct a formalism that is consistent with all known experimental data while avoiding the conceptual puzzles that have plagued quantum theory for decades. They found that the universe does not follow a single, linear history when it comes to quantum events. Instead, the history of a quantum system is a tapestry woven from pairs of trajectories, where the past and future are linked in a way that allows for interference and uncertainty without requiring a mysterious collapse. This work does not change the predictions of quantum mechanics, but it changes the story we tell about how the universe works, replacing a disjointed narrative with a unified, continuous description of reality that emerges directly from the data itself.

Technical Summary: Phenomenological Quantum Mechanics I: Phenomenology of Quantum Observables

Problem Statement
The standard formalism of quantum mechanics is characterized by a dichotomous mode of description: unitary evolution for non-measurement contexts and the Born rule with state collapse for measurement contexts. This structure necessitates the "Heisenberg cut," an arbitrary boundary between the quantum system and the classical observer, which remains an unresolved conceptual issue. Furthermore, the standard "state-centric" formulation, historically derived from single-measurement phenomenology, struggles to naturally accommodate sequential measurements and multi-time correlations without invoking the collapse postulate as an ad hoc interface. The authors propose a systematic reconstruction of quantum mechanics starting solely from phenomenological observations of sequential measurements, avoiding any presuppositions regarding underlying mathematical structures (such as Hilbert spaces) or the existence of a state vector.

Methodology
The authors employ a "phenomenological quantum mechanics" approach, treating the outcomes of sequential measurement experiments as the sole input for deducing the formalism.

Input Data: The foundational inputs are multi-time probability distributions, $P_{t_n}^{F_n}(f_n)$ , estimated from experimental data (or theoretically calculated via the standard Born rule, treated here as a proxy for empirical observation). These distributions describe sequences of results $f_n = (f_n, \dots, f_1)$ obtained from devices measuring observables $F_n, \dots, F_1$ at times $t_n > \dots > t_1$ .
Phenomenological Analysis: The paper systematically analyzes these probability distributions to identify properties that distinguish quantum systems from classical ones. Key observations include:
- Causality: Past results influence future probabilities, but future choices do not retroactively alter past distributions.
- Inconsistency: Marginalizing over intermediate measurements does not yield the probability distribution of a shorter sequence (violating the Kolmogorov consistency condition).
- Coarse-Graining: Proper quantum coarse-grained devices (which physically merge outcomes without post-processing) yield results distinct from the sum of fine-grained alternatives, demonstrating genuine quantum interference.
- Pair-wise Interference: Interference effects decompose into sums of pair-wise interactions between alternatives, rather than involving higher-order ( $k>2$ ) interference.
- Independent Systems: Interference terms between independent subsystems factorize in a specific manner involving complex co-interference terms.
- Markovianity and Zeno Effect: Fine-grained measurements exhibit Markovian properties, and the Quantum Zeno effect implies a specific short-time quadratic behavior of survival probabilities.
- Uncertainty Relations: Inequivalent fine-grained observables exist, violating classical certainty relations.

Key Contributions and Results
The paper demonstrates that these phenomenological constraints necessitate a formalism rooted in "bi-trajectories" rather than single trajectories or state vectors.

Derivation of Bi-Probability Distributions:
To account for pair-wise interference and the factorization of independent systems, the authors introduce complex-valued distributions on the space of sequence pairs, $Q_{t_n}^{F_n}(f_n^+, f_n^-)$ . These "bi-probability distributions" satisfy:
- Normalization: $\sum Q = 1$ .
- Causality: Diagonal dominance in the time evolution.
- Factorization: For independent systems, $Q_{AB} = Q_A Q_B$ .
- Positive Semi-Definiteness: Ensuring the diagonal elements correspond to valid probabilities.
- Bi-Consistency: A generalized Kolmogorov consistency condition for pairs of sequences.
The Bi-Trajectory Picture:
By combining the phenomenological observations of Markovianity and the Zeno effect (which imply uniform boundedness of the distributions), the authors invoke a generalized Kolmogorov extension theorem. This proves the existence of a complex-valued measure on the space of trajectory pairs, $Q[ f^+, f^- ][Df^+][Df^-]$ .
- The observed probability of a sequence $f_n$ is recovered as the diagonal of this measure: $P(f_n) = Q(f_n, f_n)$ .
- This establishes that quantum mechanics is fundamentally a bi-trajectory theory, where observed probabilities arise from the superposition of interfering pairs of elementary trajectories.
Resolution of the Dichotomy:
The resulting formalism eliminates the need for the Heisenberg cut and the state collapse postulate. The description is unified: the dynamics are governed by the bi-trajectory measure, and "measurement" is simply the extraction of diagonal elements from the bi-probability distribution. The formalism naturally accommodates classical limits (where the measure becomes diagonal and reduces to a single trajectory) and quantum systems without invoking a separate "collapse" rule.

Significance
This paper (Part I of a two-part series) establishes that the phenomenology of sequential measurements is rich enough to deduce a formalism based on bi-trajectories.

Non-Classical Nature: It rigorously demonstrates that quantum systems cannot be described by a single trajectory (uni-trajectory) because multi-time probabilities violate classical consistency conditions.
Structural Insight: It identifies the "bi-trajectory" structure as the natural mathematical consequence of pair-wise interference and the factorization of independent systems.
Formal Reconstruction: While this paper successfully deduces the bi-trajectory picture from phenomenology, the demonstration that this formalism is fully functional and can be reduced to the standard formulation (Hilbert spaces, density matrices, unitary evolution) is reserved for Part II of the series.
Completeness: The authors argue that the bi-trajectory elements, once brought together as a holistic formalism in the subsequent paper, will reproduce all empirically confirmed predictions of standard quantum mechanics while providing a unified description for sequential measurements without the need for an arbitrary observer-system cut.

The authors conclude that the "state-centric" view is an artifact of focusing on single measurements, whereas the "bi-trajectory" view is the necessary structure emerging from the full phenomenology of sequential probing.

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