We provide an introduction to the theory of quantum measurements that is centered on the pivotal role played by John von Neumann's model. This introduction is accessible to students and researchers from outside the field of foundations of quantum mechanics and presented within a historical context. We first explain the origins and the meaning of the measurement problem in quantum theory, and why it is not present in classical physics. We perform a chronological review of the quantization of action and explain how this led to successive restrictions on what could be measured in atomic phenomena, until the consolidation of the orthodox interpretation of quantum mechanics. The clear separation between quantum system and classical apparatus that causes these restrictions is subverted in von Neumann's paradigmatic model of quantum measurements, a subject whose concepts we explain, while also providing the mathematical tools necessary to apply it to new problems. We show how this model was important in discussing the interpretations of quantum mechanics and how it is still relevant in modern applications. In particular, we explain in detail how it can be used to describe weak measurements and the surprising results they entail. We also discuss the limitations of von Neumann's model of measurements, and explain how they can be overcome with POVMs and Kraus operators. We provide the mathematical tools necessary to work with these generalized measurements and to derive master equations from them. Finally, we demonstrate how these can be applied in research problems by calculating the Quantum Zeno Effect.
We provide an introduction to the theory of quantum measurements that is centered on the pivotal role played by John von Neumann's model. This introduction is accessible to students and researchers from outside the field of foundations of quantum mechanics and presented within a historical context. We first explain the origins and the meaning of the measurement problem in quantum theory, and why it is not present in classical physics. We perform a chronological review of the quantization of action and explain how this led to successive restrictions on what could be measured in atomic phenomena, until the consolidation of the orthodox interpretation of quantum mechanics. The clear separation between quantum system and classical apparatus that causes these restrictions is subverted in von Neumann's paradigmatic model of quantum measurements, a subject whose concepts we explain, while also providing the mathematical tools necessary to apply it to new problems. We show how this model was important in discussing the interpretations of quantum mechanics and how it is still relevant in modern applications. In particular, we explain in detail how it can be used to describe weak measurements and the surprising results they entail. We also discuss the limitations of von Neumann's model of measurements, and explain how they can be overcome with POVMs and Kraus operators. We provide the mathematical tools necessary to work with these generalized measurements and to derive master equations from them. Finally, we demonstrate how these can be applied in research problems by calculating the Quantum Zeno Effect.
We provide an introduction to the theory of quantum measurements that is centered on the pivotalrole played by John von Neumann’s model. This introduction is accessible to students andresearchers from outside the field of foundations of quantum mechanics and presented withina historical context. We first explain the origins and the meaning of the measurement problemin quantum theory, and why it is not present in classical physics. We perform a chronologicalreview of the quantization of action and explain how this led to successive restrictions onwhat could be measured in atomic phenomena, until the consolidation of the orthodox interpretationof quantum mechanics. The clear separation between quantum system and classicalapparatus that causes these restrictions is subverted in von Neumann’s paradigmatic modelof quantum measurements, a subject whose concepts we explain, while also providing themathematical tools necessary to apply it to new problems. We show how this model was importantin discussing the interpretations of quantum mechanics and how it is still relevant inmodern applications. In particular, we explain in detail how it can be used to describe weakmeasurements and the surprising results they entail. We also discuss the limitations of vonNeumann’s model of measurements, and explain how they can be overcome with POVMs andKraus operators. We provide the mathematical tools necessary to work with these generalizedmeasurements and to derive master equations from them. Finally, we demonstrate how thesecan be applied in research problems by calculating the Quantum Zeno Effect.
Abstract The book starts with a description of classical mechanics then discusses the quantum phenomena that require us to give up our commonsense classical intuitions. We consider the physical and conceptual arguments that led to the standard von Neumann-Dirac formulation of quantum mechanics and how the standard theory explains quantum phenomena. This includes a discussion of how the theory’s two dynamical laws work with the standard interpretation of states to explain determinate measurement records, quantum statistics, interference effects, entanglement, decoherence, and quantum nonlocality. A careful understanding of how the standard theory works ultimately leads to the quantum measurement problem. We consider how the measurement problem threatens the logical consistency of the standard theory then turn to a discussion of the main proposals for resolving it. This includes collapse formulations of quantum mechanics like Wigner’s extension of the standard theory and the GRW approach and no-collapse formulations like pure wave mechanics, the various many-worlds theories, and Bohmian mechanics. In discussing alternative formulations of quantum mechanics we pay particular attention to the explanatory role played by each theory’s empirical ontology and associated metaphysical commitments and the conceptual trade-offs between theoretical options.
This paper explores a thought experiment: what would physics look like if quantum mechanics had not originated in the early 20th century from the exploration of the physical properties of microscopic particles (such as blackbody radiation and the particle and wave nature of electron orbits), but had instead been established first as a dynamics and probability theory about "physical measurement" itself? Through extrapolation from this historical turning point, this paper reconstructs a theoretical framework centered on "measurement dynamics," not on "particles," "wave functions," or "quantum fields" as fundamental ontology, but on a trinity of "system-measuring instrument-observer." Research shows that under this paradigm, the "measurement problem" is no longer a philosophical dilemma added after the theory's establishment, but rather the logical starting point for its construction; wave functions and quantum fields are relegated to derivative mathematical tools for calculating measurement probability distributions. This paper elaborates on the dynamics of measurement operators, the law of information evolution, and a systematic critique of counterfactual definiteness under this paradigm, providing a new and unified perspective for re-examining the fundamental interpretation of modern quantum mechanics.
Offering a detailed account of the key concepts and mathematical apparatus of quantum mechanics, this textbook is an ideal companion to both undergraduate and graduate courses. The formal and practical aspects of the subject are explained clearly alongside examples of modern applications, providing students with the tools required to thoroughly understand the theory and apply it. The authors provide an intuitive conceptual framework that is grounded in a coherent physical explanation of quantum phenomena, established over decades of teaching and research in quantum mechanics and its foundations. The book's educational value is enhanced by the inclusion of examples and exercises, with solutions available online, and an extensive bibliography is provided. Notes throughout the text provide fascinating context on the tumultuous history of quantum mechanics, the people that developed it, and the questions that still remain at its center. This title is also available as Open Access on Cambridge Core.
We present a comprehensive technical review of quantum measurement theory, emphasizing the mathematical structures underlying the measurement problem and their role in modern quantum information science. Beginning with spectral theory and the Born rule, we develop the theory of positive operator-valued measures (POVMs), Kraus operators, and completely positive channels as essential generalizations of projective measurements. We discuss how decoherence theory addresses the preferred-basis and interference problems while leaving the outcome problem formally unresolved, motivating interpretational responses ranging from collapse models to many-worlds scenarios. Throughout, we emphasize the applicability of POVM theory to quantum state discrimination, quantum cryptography, and quantum error correction. The review prioritizes physical motivation and mathematical hints over detailed proofs, suitable for researchers transitioning into quantum foundations or quantum information theory.
The theoretical description of measurement in quantum mechanics is one of the most controversial topics in physics. It is deeply entrenched in the interpretations of quantum mechanics and combines both physics and philosophy into the discussion. Often it is described as the measurement problem because of some issues associated with the theory of measurement introduced by John von Neumann in 1932, related to the fact that the Born projection postulate is not unitary and so is not described by the time-dependent Schrödinger equation. We discuss that theory later in this chapter, but we start from a simpler approach focused on what is actually being done in most single-quantum experiments. Along the way, we encounter a number of counterintuitive and subtle issues, before we explain why we refuse to tackle the measurement problem itself. Specific experiments that we discuss in detail include how we can measure individual quanta and how we can manipulate entanglement and perform delayed choice experiments, quantum beats, atom interferometry, and quantum non-demolition experiments in cavity QED.