Einstein's article on the EPR paradox is the most cited of his works, but not many know that it was not fully representative of the way he thought about the incompleteness of the quantum formalism. Indeed, his main worry was not Heisenberg's uncertainty principle, which he accepted, but the experimental non-separability of spatially separate systems. The same problem was also recognized, years later, by one of us, as part of an axiomatic analysis of the quantum formalism, which revealed an unexpected structural limitation of the quantum formalism in Hilbert space, preventing the description of separate systems. As we will explain, this limitation does not manifest at the level of the states, but of the projectors describing the properties, in the sense that there are not enough properties in the formalism to describe separate systems. The question remains whether separability is a possibility at the fundamental level and if a formalism should integrate it into its mathematical structure, as a possibility. To aid our intuition, we offer a reflection based on a powerful analogy between physical systems and human conceptual entities, as the question of separability also arises for the latter.
Einstein's article on the EPR paradox is the most cited of his works, but not many know that it was not fully representative of the way he thought about the incompleteness of the quantum formalism. Indeed, his main worry was not Heisenberg's uncertainty principle, which he accepted, but the experimental non-separability of spatially separate systems. The same problem was also recognized, years later, by one of us, as part of an axiomatic analysis of the quantum formalism, which revealed an unexpected structural limitation of the quantum formalism in Hilbert space, preventing the description of separate systems. As we will explain, this limitation does not manifest at the level of the states, but of the projectors describing the properties, in the sense that there are not enough properties in the formalism to describe separate systems. The question remains whether separability is a possibility at the fundamental level and if a formalism should integrate it into its mathematical structure, as a possibility. To aid our intuition, we offer a reflection based on a powerful analogy between physical systems and human conceptual entities, as the question of separability also arises for the latter.
Abstract Einstein’s article on the Einstein-Podolsky-Rosen paradox is the most cited of his works, but not many know that it was not fully representative of the way he thought about the incompleteness of the quantum formalism. Indeed, his main worry was not Heisenberg’s uncertainty principle, which he accepted, but the experimental non-separability of spatially separate systems. The same problem was also recognized, years later, by one of us, as part of an axiomatic analysis of the quantum formalism, which revealed an unexpected structural limitation of the quantum formalism in Hilbert space, preventing the description of separate systems. As we will explain, this limitation does not manifest at the level of the states, but of the projectors describing the properties, in the sense that there are not enough properties in the formalism to describe separate systems. The question remains whether separability is a possibility at the fundamental level and if a formalism should integrate it into its mathematical structure, as a possibility. To aid our intuition, we offer a reflection based on a powerful analogy between physical systems and human conceptual entities, as the question of separability also arises for the latter. This article is part of the theme issue ‘Newton, Principia, Newton Geneva Edition (17th–19th) and modern Newtonian mechanics: heritage, past & present’.
The measurement problem in quantum mechanics has the character of a fundamental incompleteness within that theory similar to the incompleteness of axiomatic systems in mathematics, discovered and elaborated by Gödel and others. The difficulty of describing the measurement process by the time-dependent Schrodinger equation may reflect the limitations of formal language , and quantum theory may thus require a formalism consisting of two levels of description, one for the dynamics and one for measurement, levels whose relationship resembles that of a calculus and meta-calculus.
The Einstein–Podolsky–Rosen paradox and Bell’s theorem sit at the center of one of the most famous unresolved questions in physics: how can distant quantum outcomes remain so perfectly correlated without reducing reality to either hidden variables or some troubling form of nonlocality? For nearly a century, this debate has shaped the foundations of quantum theory, the interpretation of entanglement, and, more recently, the conceptual language surrounding quantum information and quantum computing. Yet one possibility may still be underexplored: what if the puzzle begins from the wrong ontological starting point? This paper asks exactly that question. Standard quantum mechanics already accepts many systems as physically unified wholes described by a single composite wavefunction: atoms, multi-electron states, molecular orbitals, and hadrons. Even increasingly large molecules continue to exhibit interference, showing that quantum oneness is not destroyed simply by size or spatial extension. If such preserved composite oneness is already an accepted part of quantum theory, then why is the EPR–Bell pair so often framed first as two independently sufficient distant objects? The proposal developed here is both simple and provocative: the EPR–Bell pair may, prior to measurement, belong to the same ontological class as other accepted composite quantum systems. On this view, the pair is not first two complete distant realities that must somehow remain coordinated across space, but one preserved composite wavefunction whose support spans separated regions and whose distinct local outcomes emerge only when the whole state is resolved. If that is right, then one of the deepest questions in modern physics changes form. The issue is no longer simply how two distant objects stay correlated. The issue becomes whether the assumption of separability was imposed too early. This paper develops that conjecture step by step, using accepted quantum examples, the original EPR question, Bell’s constraint, and the logic of composite Wavefunction Oneness. The result is a new way of reopening the nonlocality problem—not by denying Bell, and not by proposing hidden variables, but by asking whether the EPR–Bell discussion has been starting from the wrong picture of what the physical system actually is.
We investigate the question of 'why customary macroscopic entities appear to us humans as they do, i.e., as bounded entities occupying space and persisting through time', starting from our knowledge of quantum theory, how it affects the behavior of such customary macroscopic entities, and how it influences our perception of them. For this purpose, we approach the question from three perspectives. Firstly, we look at the situation from the standard quantum angle, more specifically the de Broglie wavelength analysis of the behavior of macroscopic entities, indicate how a problem with spin and identity arises, and illustrate how both play a fundamental role in well-established experimental quantum-macroscopical phenomena, such as Bose-Einstein condensates. Secondly, we analyze how the question is influenced by our result in axiomatic quantum theory, which proves that standard quantum theory is structurally incapable of describing separated entities. Thirdly, we put forward our new 'conceptual quantum interpretation', including a highly detailed reformulation of the question to confront the new insights and views that arise with the foregoing analysis. At the end of the final section, a nuanced answer is given that can be summarized as follows. The specific and very classical perception of human seeing-light as a geometric theory-and human touching-only ruled by Pauli's exclusion principle-plays a role in our perception of macroscopic entities as ontologically stable entities in space. To ascertain quantum behavior in such macroscopic entities, we will need measuring apparatuses capable of its detection. Future experimental research will have to show if sharp quantum effects-as they occur in smaller entities-appear to be ontological aspects of customary macroscopic entities. It remains a possibility that standard quantum theory is an incomplete theory, and hence incapable of coping ultimately with separated entities, meaning that a more general theory will be needed.
In Mathematical Foundations of Quantum Mechanics (1932) von Neumann made separability of Hilbert space an axiom. Subsequent work in mathematics (some of it by von Neumann himself) investigated non-separable Hilbert spaces, and mathematical physicists have sometimes made use of them. This note discusses some of the problems that arise in trying to treat quantum systems with non-separable spaces. Some of the problems are "merely technical" but others point to interesting foundations issues for quantum theory, both in its abstract mathematical form and its applications to physical systems. Nothing new or original is attempted here. Rather this note aims to bring into focus some issues that have for too long remained on the edge of consciousness for philosophers of physics.
When developing and operating quantum computers, it is necessary to be able to check quantum registers for the presence of quantum entanglement between its subsystems, due to which quantum computers gain computational advantage in solving many problems over classical computers. The state in which a quantum system consists of two or more subsystems that are not entangled with each other is called separable.In this article, we provide an overview of various existing approaches to solving the problem of determining separability in a quantum system. To do this, we provide an overview of works on this topic, describing both established and new approaches, highlighting directions in which new possibilities for solving the problem can be explored. We provide a definition of the problem of checking a system for separability, based on which we describe several applicable separability criteria. We provide a model-theoretic approach to solving the separability problem, describing a quantum system as a blurry model.