Quantum states are the key mathematical objects in quantum theory. It is
therefore surprising that physicists have been unable to agree on what a
quantum state represents. There are at least two opposing schools of thought,
each almost as old as quantum theory itself. One is that a pure state is a
physical property of system, much like position and momentum in classical
mechanics. Another is that even a pure state has only a statistical
significance, akin to a probability distribution in statistical mechanics. Here
we show that, given only very mild assumptions, the statistical interpretation
of the quantum state is inconsistent with the predictions of quantum theory.
This result holds even in the presence of small amounts of experimental noise,
and is therefore amenable to experimental test using present or near-future
technology. If the predictions of quantum theory are confirmed, such a test
would show that distinct quantum states must correspond to physically distinct
states of reality.
Quantum states are the key mathematical objects in quantum theory. It is
therefore surprising that physicists have been unable to agree on what a
quantum state represents. There are at least two opposing schools of thought,
each almost as old as quantum theory itself. One is that a pure state is a
physical property of system, much like position and momentum in classical
mechanics. Another is that even a pure state has only a statistical
significance, akin to a probability distribution in statistical mechanics. Here
we show that, given only very mild assumptions, the statistical interpretation
of the quantum state is inconsistent with the predictions of quantum theory.
This result holds even in the presence of small amounts of experimental noise,
and is therefore amenable to experimental test using present or near-future
technology. If the predictions of quantum theory are confirmed, such a test
would show that distinct quantum states must correspond to physically distinct
states of reality.
A century after the development of quantum theory, the interpretation of a quantum state is still discussed. If a physicist claims to have produced a system with a particular quantum state vector, does this represent directly a physical property of the system, or is the state vector merely a summary of the physicist's information about the system? Assume that a state vector corresponds to a probability distribution over possible values of an unknown physical or 'ontic' state. Then, a recent no-go theorem shows that distinct state vectors with overlapping distributions lead to predictions different from quantum theory. We report an experimental test of these predictions using trapped ions. Within experimental error, the results confirm quantum theory. We analyse which kinds of models are ruled out.
Even after a hundred years of quantum theory, the true meaning of the quantum state has not yet been clearly understood. Does a pure quantum state, described by a wave function φ, represent a real physical quantity, comparable to position and momentum in classical mechanics, or does it, even as a pure state, only have a statistical significance, similar to a probability distribution? In this paper, we show that there is no contradiction between these two interpretations. Two concepts must be distinguished here: on the one hand, physical objects, which have a finite lifetime, and on the other hand, subsystems (i.e., tensor factors) of the Hilbert space of the universe. A complete description of an object at a point in time t is given by a linear subspace A of its Hilbert space, which, in the case of dim A = 1, can be specified by a wave function φ as A = [φ]. However, for every subsystem Q, for each point in time t, and for every condition B, the conditional quantum state W = W(Q,t|B) can be defined as a statistical operator on the Hilbert space of Q. Using W, certain (conditional) probabilities can be calculated. In the case of rank W = 1, the operator W can also be specified by a wave function φ, namely as W = |φ><φ|. It is the subsystems of the Hilbert space that play the role of "quantum systems" in the Copenhagen formalism, and the quantum states defined for them behave exactly as described by this formalism. If a wave function φ represents the description of a (quantum) object, then we are dealing with a real physical property, that is, φ plays an "ontic" role. However, if φ represents a quantum state, then its role is merely "statistical". If, in this case, condition B corresponds to the knowledge of a subject, φ plays an "epistemic" role. Additional Remarks The reasoning in this paper is largely based on a new systematic approach to solving the interpretation problem of quantum theory, which was originally published in 2018 (in German) under the title "Realistische Quantentheorie". A comprehensive account of this approach (in English) was published in February 2026 under the title "Realistic Quantum Theory: Solving the Interpretation Problem". It is available in the Zenodo repository (zenodo.18644694, see link below under "Related works" / "References"). The present document provides an English version of an earlier paper in German under the same title, which is also available in the Zenodo repository (zenodo.10694813, see link below).
Quantum Mechanics and its Interpretations Peter E. Hodgson (bio) Quantum Mechanics is an elegant and indeed beautiful formalism, and enables the results of many physical measurements to be calculated with great accuracy. But from the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything, it is becoming more intense over the years. It should therefore be evident that the most that can be done here is to try to show where the real difficul-ties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. The Discovery of the Quantum The quantum nature of the physical world was discovered by Max Planck in the early years of the twentieth century. He was studying thermodynamics and his attention was drawn to the frequency distribution of black-body radiation. This radiation is emitted from a small hole in a heated enclosure with rough walls, which ensures [End Page 62] that the radiation reaches statistical equilibrium before emission. The frequency distribution is found to be independent of the material lining the walls of the oven, so Planck realized that it must be a fundamental characteristic of all matter. This frequency distribution had been measured by several experimentalists, and Planck resolved to try to see how it can be derived theoretically. Already Rayleigh and Jeans had used classical radiation theory to derive an expression that fitted the data well at low frequencies but tended to infinity at high energies, which was of course unacceptable. The physicist Wien had derived a formula that fitted the data at high frequencies but failed at low frequencies. Planck used an ingenious argument based on entropy to interpolate between these two formulae and obtained a formula, now known as the Planck distribution, that fitted the data perfectly. His next task was to understand the physics behind his formula. He used the familiar technique of assuming that the energy is emitted in small finite amounts, and then let their size tend to zero, so as to represent a continuous distribution. To his great surprise, the intermediate formula with discrete amounts of energy, called quanta, gave the Planck distribution, but the final result, with infinitely small quanta, did not. All his instincts as a physicist rebelled against this conclusion. Surely energy must be emitted continuously. He tried hard to find a way to avoid this unwelcome conclusion, but without success. He had to admit that it is a feature of the world. Soon it was confirmed by Einstein’s interpretation of the photoelectric effect. This story shows clearly that physicists do not impose their ideas on the world, but recognize and accept what they find, whether or not it agrees with their original ideas. The world is made by God in ways that often we cannot imagine. [End Page 63] The Old Quantum Theory A few years later Rutherford discovered that the atom has a small central nucleus containing most of its mass surrounded by a number of electrons. Bohr used the idea of the quantum to build his model of the atom. He assumed that the electrons go around the nucleus in discrete orbits, and that radiation is emitted when an electron falls from one orbit to a lower one. With this model, he was able to calculate to high accuracy the frequencies of the radiation emitted from an excited hydrogen atom. This was a great achievement, but to get his result he had to assume that the electrons stayed in their orbits, whereas classical theory predicted that they would continuously lose energy and rapidly spiral into the nucleus. Quantum Mechanics and its Interpretations These problems were solved by the development of quantum mechanics in the 1920s by Heisenberg, Schrodinger, Dirac, and others. This enabled what is called the wave function to be calculated for any physical system, and from the wave function the probabilities of occurrence of measurable quantities can be found. It is a beautiful theory, and is the indispensable means of calculation for atomic, nuclear...
Quantum theory is a phenomenally successful cornerstone of physics. But a recent poll among quantum physicists shows that there is still little agreement on what the theory tells us about physical reality. Yet precious bits of consensus are emerging. As always, Albert Einstein was ahead of his time. Quantum theory was still in its infancy, its formalism a mere sketch, and its implications only beginning to be realized. Yet Einstein, in 1917, already spotted what would trouble him throughout his lifetime. He concluded that the nascent theory, which he had helped initiate, seemed to leave “time and direction of elementary processes,” such as the spontaneous emission of radiation from a molecule, “to chance” 1. Einstein's conclusion was visionary. It was also incredibly bold, for it suggested a radical departure from the powerful, centuries-old deterministic and causal worldview of classical physics. But Einstein also immediately expressed his discomfort with the idea of fundamental chance. In fact, his worries were just the seed of what would turn into one of the longest and most heated debates in the history of physics. What does quantum theory mean, and what does it tell us about the nature of reality? Some of the brightest minds in physics locked horns over these issues. Besides Einstein, there were Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Pascual Jordan, Carl Friedrich von Weizsäcker, Wolfgang Pauli, and Richard Feynman, just to name a few. Ask anyone today working on foundational questions in quantum theory and you are likely to hear that there is still no consensus on many of these questions—all the while, of course, everybody seems to be in perfect agreement on how to apply the quantum formalism when it comes to making experimental predictions. Remarkably, there has never been any serious attempt to find out what the current opinions of the community really are and how they may be correlated. We jumped on the opportunity and polled 33 participants at a 2011 conference on “Quantum Physics and the Nature of Reality” (Fig. 1) 2. (We hasten to add that our poll is to be understood as merely a snapshot and certainly cannot claim to be representative.) As soon as we had collected the questionnaires, the respondents eagerly asked to hear the results. Evidently, we had not been the only curious party. What did we find? The fundamental questions, it seems, still elicit radically different responses. Take the quantum state—represented by the wavefunction—as an illustration. The wavefunction is the central mathematical symbol of quantum mechanics. It allows us to make incredibly accurate predictions, be it in atomic, molecular, or solid-state physics. But how are we to interpret it? Is it itself a physical property, or does it merely represent our knowledge? Our poll revealed a tie between these two positions. And not only that: a third of respondents preferred a mix of both viewpoints—perhaps a sign that a more nuanced view, or even a synthesis of views, may be on the horizon. Only 3% of those polled thought that quantum states never refer to a single system but only to ensembles of systems. Perhaps this departure from the ensemble view—a view that, we suspect, remains more popular in the physics community at large—has to do with the fact that experiments with single quantum systems have become routine in today's quantum-information laboratories. Sharp disagreement was in evidence on another deep question. Abraham Pais, Einstein's biographer, recalled that “during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it” 3. Putting the issue differently, are the physical properties of objects well defined before we actually measure them? Half of the respondents in our poll believed that they sometimes were, while the other half said that they never were. These two positions correspond to radically different ways of understanding the function of measurement: either as simply ascertaining—at least sometimes—of what is already there, or as an act of genuine creation. The role of measurement is also at the heart of another perennial favorite of foundational discussions, the measurement problem. The measurement problem highlights a curious fact about quantum theory: while the act of measurement assumes a fundamental role, it is not described—not without interpretive contortions, anyway—by the dynamical law of quantum theory, the Schrödinger equation. For decades, the measurement problem was seen by many as the main stumbling block for quantum theory, sparking a storm of rivaling attempts to deal with this perceived issue. Today, the waters seem to have calmed and attention shifted elsewhere, as only less than a fourth of respondents consider the measurement problem “a severe difficulty threatening quantum mechanics.” Changing attitudes are also evident when it comes to the issue of chance Einstein had brought to light in 1917. Today, the question of the randomness of quantum events no longer seems to be all that contentious. Most of our respondents accepted randomness as irreducible or a fundamental concept in nature (or both). Less than one-tenth of respondents thought that quantum randomness was merely apparent rather than fundamental, and no one believed in a hidden determinism governing quantum events. We wish we could hear Einstein's opinion today. Why have attitudes and priorities changed? It's hard to say, but we think there may be several driving forces. The rapid ascent of quantum information science has not only led to applications such as secure cryptography and the promise of lightning-fast quantum computers, but has also shed new light on foundational problems and handed us new tools for addressing them. Our poll shows that these tools are considered a welcome addition, with three-fourths seeing quantum information as a breath of fresh air for foundational investigations. There is also optimism about the future technological development of quantum information, with more than 75% of respondents expecting a working and useful quantum computer to be realized within the next fifty years. Such optimism might be rooted in the astonishing experimental progress we have witnessed over the past decades, another driving force behind changing attitudes. We are now at stage where almost perfect experimental control of single quantum systems has been achieved. The 2012 Nobel Prize in physics was awarded for just such experiments. This has put at our fingertips the ability to actually carry out the thought experiments—or approximations thereof, at any rate—that the founders of quantum theory had dreamed up. Take Schrödinger's cat. In 1935, Schrödinger wanted to illustrate what seemed to him an absurd prediction of the new theory: that the strange quantum superpositions of the atomic world can be amplified to everyday objects. Schrödinger's object of choice was a cat in a specifically prepared box. Quantum theory then suggested that the state of the cat should become transformed into a bizarre quantum superposition of being simultaneously alive and dead. But what Schrödinger had intended as a “burlesque” 4 thought experiment is now approaching experimental reality. Over the last decades, researchers have become increasingly skilled in breeding small versions of Schrödinger's cat. While they don't involve actual cats, they do already involve large biological molecules. Our respondents, it seems, were inspired by such developments. A solid two-thirds majority declared that they don't expect any fundamental limit in this enterprise of observing larger and larger objects in superpositions of macroscopically distinct states. No limit, then, for quantum theory—something that Bohr had already suggested when he, in some of his early thought experiments, considered apparatuses that included macroscopic parts behaving in quantum-mechanical ways. There are other new trends. Why do people seem to be much more at ease with relativity theory than with quantum theory? For one, both special and general relativity theory are grounded in a few simple physical principles. By contrast, quantum theory is lacking such a foundation and is instead based on a rather abstract mathematical body. Today, several researchers are out to change this situation, trying to derive quantum theory from a small set of physical or information-theoretic principles. They hope that in this way, we will understand why we have this peculiar theory to begin with. A majority in our poll regarded such “reconstructions” of quantum theory as giving useful insights, indicating once more that any new tool that promises fresh insights is embraced with an open mind. To be sure, many (albeit not all) of the questions we asked in the poll are interpretive questions. This means that unless we should find out some day that quantum theory is in fact not always valid—in which case we would have to rethink the whole game—experiments arguably can't play the role of the arbiter. Some physicists will conclude from this that we need not bother with such questions, and relegate them to philosophy. We think this is too quick. Many ideas in physics didn't come out of thin air, but came about when someone started looking at a problem with a certain philosophical disposition. In this sense, we may think of the plurality of views in quantum mechanics as productive. Every interpretation highlights some aspect of quantum theory, is particularly suited for explaining particular phenomena, and may inspire new technologies—such as the quantum computer—or even new physics. What, then, does the future hold? Half of the respondents in our poll believe that in fifty years from now, we will probably still be coming together for conferences on the fundamental questions of quantum theory. But what kinds of questions will be on the agenda? Will we still be worrying about the same problems as today? Or will these problems have been solved—or found to be less relevant—and our focus shifted to new questions once again? Who knows. But the location of the conference where we conducted the poll may tell a story. It was the place of a monastery, and a central position in the battle between Catholics and Protestants during the seventeenth century. Many of the questions that were hotly debated back then are no longer of interest to many people today, and other, new questions have since taken center stage in our society. Maybe some of the battles over the interpretation of quantum theory will meet a similar fate.
This year we celebrate 100 years of quantum mechanics (QM). Incorrect interpretations of QM and incorrect mental models of the invisible details of quantum phenomena lead to paradoxes. To explain these, we advocate the statistical contextual interpretation (SCI) of quantum mechanics. State vectors (wave functions) and various operators are purely mathematical entities that permit quantitative probabilistic predictions. “State vector” describes an ensemble of identically prepared physical systems, and a specific “operator” represents a class of equivalent measurements of a physical observable. A collapse of wavefunction is not a mysterious and instantaneous physical process; a collapsed quantum state describes a new ensemble of physical systems prepared in a particular way. A value of a physical observable, such as a spin projection, associated with a pure quantum ensemble is a characteristic of this ensemble created by its interaction with measuring instruments. Probabilities are objective properties of random experiments in which empirical frequencies stabilize. Following Einstein, SCI rejects the claim that QM provides a complete description of individual physical systems, but it remains agnostic about whether a more detailed subquantum description can be found or is necessary. In conformity with Bohr contextuality, SCI rejects Bell-local and Bell-causal hidden variable models. Nevertheless, by incorporating into a probabilistic model contextual hidden variable measuring instruments, long distance quantum correlations studied in Bell tests can be explained without evoking quantum nonlocality or retro-causality. SCI allows the explanation of several quantum phenomena without evoking quantum magic. SCI does not claim to provide a complete description of quantum phenomena; in fact, it is unknown whether quantum probabilities even provide a complete description of existing experimental data. Time series of experimental data may contain much more information than is obtained using empirical frequencies and histograms. Therefore, predictable completeness of QM must be tested and not taken for granted.
Despite the unparalleled accuracy of quantum-theoretical predictions across an enormous range of phenomena, the theory's foundations are still in doubt. The theory deviates radically from classical physics, predicts counterintuitive phenomena, and seems inconsistent. The biggest stumbling block is measurement, where the Schrodinger equation's unitary evolution seems inconsistent with collapse. These doubts have inspired a variety of proposed interpretations and alterations of the theory. Most interpretations posit the theory represents only observed appearances rather than reality. The realistic interpretations, on the other hand, posit entities such as other universes, hidden variables, artificial collapse mechanisms, or human minds, that are not found in the standard mathematical formulation. Surprisingly, little attention has been paid to the possibility that the standard theory is both realistic and correct as it stands. This paper examines several controversial issues, namely quantization, field particle duality, quantum randomness, superposition, entanglement, non-locality, and measurement, to argue that standard quantum physics, realistically interpreted, is consistent with all of them.